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1 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg

2 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 1 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 3 Two or more random variables 3.1 Joint probability distribution function 3.2 The discrete case: Joint probability mass function A two-dimensional random walk 3.3 The continuous case: Joint probability density function Meeting times 4 Conditional distribution and independence 5 Expectations and moments 5.1 Mean, median and mode A one-dimensional random walk 5.2 Central moments, variance and standard deviation 5.3 Moment generating functions

3 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Functions of random variables 6.1 Functions of one random variable 6.2 Functions of two or more random variables 6.3 Two or more random variables: multivariate moments 7 Inequalities 7.1 Jensen inequality 7.2 Markov s inequality 7.3 Chebyshev s inequality 7.4 Cantelli s inequality 7.5 The law of large numbers

4 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Joint probability distribution functions The joint probability distribution function of random variables X and Y, denoted by, is defined by for all x, y As before, some obvious properties follow from this definition of joint cumulative distribution function:, and are called marginal distribution functions of X and Y, resp.

5 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 4 Copulas Consider a random vector and suppose that its margins and are continuous. By applying the probability integral transformation to each component, the random vector has uniform margins. The copula of is defined as the joint cumulative distribution function of :

6 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b The discrete case: joint probability mass functions Let X and Y be two discrete random variables that assume at most a countable infinite number of value pairs, i,j = 1,2,, with nonzero probabilities. Then the joint probability mass function of X and Y is defined by for all x and y. It is zero everywhere except at the points, i,j = 1,2,, where it takes values equal to the joint probability. (Example of a simplified random walk)

7 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b The continuous case: joint probability density functions The joint probability density function of 2 continuous random variables X and Y is defined by the partial derivative Since is monotone non-decreasing in both x and y, the associated joint probability density function is nonnegative for all x and y. As a direct consequence: where are now called the marginal density functions of X and Y respectively

8 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 7 Also, and for

9 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 8 Meeting times A boy and a girl plan to meet at a certain place between 9am and 10am, each not wanting to wait more than 10 minutes for the other. If all times of arrival within the hour are equally likely for each person, and if their times of arrival are independent, find the probability that they will meet. Answer: for a single continuous random variable X that takes all values over an interval a to b with equal likelihood, the distribution is called a uniform distribution and its density function has the form

10 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 9 random variables is a flat surface within prescribed bounds. The volume under the surface is unity. The joint density function of two independent uniformly distributed

11 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 10 We can derive from the joint probability, the joint probability distribution function, as usual From this we can again derive the marginal probability density functions, which clearly satisfy the earlier definition for 2 random variables that are uniformly distributed over the interval [0,60]

12 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Conditional distribution and independence The concepts of conditional probability and independence introduced before also play an important role in the context of random variables The conditional distribution of a random variable X, given that another random variable Y has taken a value y, is defined by When a random variable X is discrete, the definition of conditional mass function of X given Y=y is For a continuous random variable X, the conditional density function of X given Y=y is

13 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 12 In the discrete case, using the definition of conditional probability, we have an expression which is very useful in practice when wishing to derive joint probability mass functions Using the definition of independent events in probability theory, when the random variables X and Y are assumed to be independent, so that

14 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 13 The definition of a conditional density function for a random continuous variable X, given Y=y, entirely agrees with intuition : By setting and by taking the limit, this reduced to provided

15 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 14 From and we can derive that a form that is identical to the discrete case. But note that

16 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 15 When random variables X and Y are independent, however, (using the definition for ) and (using the expression it follows that

17 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 16 Finally, when random variables X and Y are discrete, and in the case of a continuous random variable, Note that these are very similar to those relating the distribution and density functions in the univariate case. Generalization to more than two variables should now be straightforward, starting from the probability expression

18 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 17 Resistor problem Resistors are designed to have a resistance of R of. Owing to some imprecision in the manufacturing process, the actual density function of R has the form shown (right), by the solid curve. Determine the density function of R after screening (that is: after all the resistors with resistances beyond the range are rejected. Answer: we are interested in the conditional density function where A is the event

19 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 18 We start by considering

20 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 19 Where is a constant. The desired function is then obtained by differentiation. We thus obtain Now, look again at a graphical representation of this function. What do you observe?

21 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 20 Answer: The effect of screening is essentially a truncation of the tails of the distribution beyond the allowable limits. This is accompanied by an adjustment within the limits by a multiplicative factor 1/c so that the area under the curve is again equal to 1.

22 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Expectations and moments 5.1 Mean, median and mode Expectations Let g(x) be a real-valued function of a random variable X. The mathematical expectation or simply expectation of g(x) is denoted by E(g(X)) and defined as if X is discrete where are possible values assumed by X. When the range of i extends from 1 to infinity, the sum above exists if it converges absolutely; that is,

23 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 22 If the random variable X is continuous, then if the improper integral is absolutely convergent, that is, then this number will exist.

24 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 23 Basic properties of the expectation operator E(.), for any constant c and any functions g(x) and h(x) for which expectations exist include: Proofs are easy. For example, in the 3 rd scenario and continuous case :

25 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 24 Two other measures of centrality of a random variable: o A median of X is any point that divides the mass of its distribution into two equal parts think about our quantile discussion o A mode is any value of X corresponding to a peak in its mass function or density function

26 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 25 From left to right: positively skewed, negatively skewed, symmetrical distributions

27 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 26 Moments of a single random variable Let ; the expectation, when it exists, is called the nth moment of X and denoted by : The first moment of X is also called the mean, expectation, average value of X and is a measure of centrality

28 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 27 A one-dimensional random walk read at home An elementary example of a random walk is the random walk on the integer number line, which starts at 0 and at each step moves +1 or 1 with equal probability. This walk can be illustrated as follows: A marker is placed at zero on the number line and a fair coin is flipped. If it lands on heads, the marker is moved one unit to the right. If it lands on tails, the marker is moved one unit to the left. After five flips, it is possible to have landed on 1, 1, 3, 3, 5, or 5. With five flips, three heads and two tails, in any order, will land on 1. There are 10 ways of landing on 1 or 1 (by flipping three tails and two heads), 5 ways of landing on 3 (by flipping four heads and one tail), 5 ways of landing on 3 (by flipping four tails and one head), 1 way of landing on 5 (by flipping five heads), and 1 way of landing on 5 (by flipping five tails).

29 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 28 Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis).

30 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 29 See the figure below for an illustration of the possible outcomes of 5 flips. To define this walk formerly, take independent random variables where each variable is either 1 or -1 with a 50% probability for either value, and set and. The series is called the simple random walk on. This series of 1 s and -1 s gives the distance walked, if each part of the walk is of length 1.

31 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 30 The expectation of is 0. That is, the mean of all coin flips approaches zero as the number of flips increase. This also follows by the finite additivity property of expectations:. A similar calculation, using independence of random variables and the fact that shows that. This hints that, the expected translation distance after n steps, should be of the order of.

32 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 31 Suppose we draw a line some distance from the origin of the walk. How many times will the random walk cross the line?

33 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 32 The following, perhaps surprising, theorem is the answer: for any random walk in one dimension, every point in the domain will almost surely be crossed an infinite number of times. [In two dimensions, this is equivalent to the statement that any line will be crossed an infinite number of times.] This problem has many names: the level-crossing problem, the recurrence problem or the gambler's ruin problem. The source of the last name is as follows: if you are a gambler with a finite amount of money playing a fair game against a bank with an infinite amount of money, you will surely lose. The amount of money you have will perform a random walk, and it will almost surely, at some time, reach 0 and the game will be over.

34 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 33 At zero flips, the only possibility will be to remain at zero. At one turn, you can move either to the left or the right of zero: there is one chance of landing on -1 or one chance of landing on 1. At two turns, you examine the turns from before. If you had been at 1, you could move to 2 or back to zero. If you had been at -1, you could move to -2 or back to zero. So, f.i. there are two chances of landing on zero, and one chance of landing on 2. If you continue the analysis of probabilities, you can see Pascal's triangle

35 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Variance and standard deviation Central moments The central moments of a random variable X are the moments of X with respect to its mean. So the nt central moment of X, denoted as, is defined as The variance of X is the second central moment and usually denoted as or Var(X). It is the most common measure of dispersion of a distribution about its mean, and by definition always nonnegative. Important properties of the variance of a random variable X include:

36 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 35 The standard deviation of X, another such measure of dispersion, is the square root of Var(X) and often denoted by. One of the advantages of using instead of is that the standard deviation of X has the same unit as the mean. It can therefore be compared with the mean on the same scale to gain some measure of the degree of spread of the distribution. A dimensionless number that characterizes dispersion relative to the mean which also facilitates comparison among random variables of different units is the coefficient of variation, defined by

37 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 36 Relation between variance and simple moments Indeed, with : Hence, there are two ways of computing variances, o using the original definition, or o using the relation to the first and second simple moments

38 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 37 Third central moment The third moment about the mean is sometimes called a measure of asymmetry, or skewness The skewness for a normal distribution is zero, and any symmetric data should have a skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. Some measurements have a lower bound and are thus skewed right. For example, in reliability studies, failure times cannot be negative.

39 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 38 Knowledge of the third moment hardly gives a clue about the shape of the distribution (e.g., f 3 (x) is far from symmetrical but its third moment is zero ) The ratio is called the coefficient of skewness and is unitless.

40 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 39

41 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 40 Fourth central moment The fourth moment about the mean is sometimes called a measure of excess, or kurtosis It refers to the degree of flatness of a density near its center, and usually the coefficient of excess kurtosis is considered (-3 ensures that the excess is zero for normal distributions): A distribution with negative excess kurtosis is called platykurtic. A distribution with positive excess kurtosis is called leptokurtic. Distributions with zero excess kurtosis are called mesokurtic This measure however suffers from the same failing as does the measure of skewness: It does not always measure what it is supposed to.

42 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 41 The importance of moments or not? In applied statistics, the first two moments are obviously of great importance. It is usually necessary to know at least the location of the distribution and to have some idea about its dispersion or spread These characteristics can be estimated by examining a sample drawn from a set of objects known to have the distribution in question (see future chapters) In some cases, if the moments are known, then the density can be determined (e.g., cfr Normal Distribution). It would be useful if a function could be found that would give a representation of all the moments. Such a function is called a moment generating function.

43 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Moment generating functions

44 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 43 Origin of the name moment generating function

45 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 44 Importance of moment generating functions In principle it is possible that there exists a sequence of moments for which there is a large collection of different distributions functions having these same moments so a sequence of moments does not determine uniquely the corresponding distribution function

46 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 45 For example:

47 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 46 Densities for two distributions with the SAME infinite series of moments

48 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 47 Is there any moment criterion for identifying distributions that would ensure that two distributions are identical? Yes: If random variables X and Y both have moment generating functions that exist in some neighborhood of zero and if they are equal for all t in this neighborhood, then X and Y have the same distributions! Simple proof of a special case:

49 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 48 Now

50 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 49

51 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Functions of random variables 6.1 Functions of one random variable Real-life examples often present themselves with far more complex density functions that the one described so far. In many cases the random variable of interest is a function of one that we know better, or for which we are better able to describe its density or distributional properties For this reason, we devote an entire part on densities of functions of random variables. We first assume a random variable X and Y=g(X), with g(x) a continuous function of X. o How does the corresponding distribution for Y look like? o What are its moment properties?

52 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 51 Discrete random variables Suppose that the possible values taken by X can be enumerated as. Then the corresponding possible values of Y can be enumerated as. Let the probability mass function of X be given by then the probability mass function of Y is determined as

53 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 52 Continuous random variables To carry out similar mapping steps as outlined for discrete random variables, care must be exercised in choosing appropriate corresponding regions in ranges spaces For strictly monotone increasing functions of x : By differentiating both sides: In general, for X a continuous random variable and Y=g(X), with g(x) continuous in X and strictly monotone,

54 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 53 Let X be a continuous random variable and Y=g(X), where g(x) is continuous in X, and y=g(x) admits at most a countable number (r) of roots then

55 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 54 Example of Y= and X normally distributed, leading to -distribution

56 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 55 Recall: the probability integral transform

57 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 56 Two ways to compute expectations

58 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 57 Two ways to compute moments Expressing the nth moment of Y as, Alternatively, using characteristic functions (here: j is the imaginary unit): after which the moments of Y are given via taking derivatives: Note that

59 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Functions of two or more random variables: sums of random variables Deriving moments mean and variances

60 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 59

61 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 60 Covariances and correlations

62 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 61 Several sets of (X, Y) ) points, with the correlation coefficient of X and Y for each set, are shown in the following plot. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). Remark: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero

63 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 62

64 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 63

65 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 64

66 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 65 The Central Limit Theorem One of the most important theorems of probability theory is the Central Limit Theorem. It gives an approximate distribution of an average.

67 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 66 Remark : In the light of the central limit theorem, our results concerning 1- dimensional random walks is of no surprise: as the number of steps increases, it is expected that position of the particle becomes normally distributed in the limit.

68 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 67 Determine the distribution :(1) Cumulative-distribution function technique

69 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 68

70 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 69 (2) Moment-generating-function technique

71 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 70

72 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 71 Multivariate transform associated with X and Y The multivariate transform of X and Y is given by It is a direct generalization of the moment generating functions we have seen for a single random variable or a sum of independent random variables: o If X and Y are independent random variables, and, then The function is called the joint moment generating function of X and Y

73 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 72 (3) The transformation technique when the number of variables grows

74 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 73

75 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 74

76 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 75

77 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 76 where x 0 is the location parameter, specifying the location of the peak of the Cauchy distribution, and γ is the scale parameter (note: mean and standard deviation are undefined!)

78 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Two or more random variables: multivariate moments Let X 1 and X 2 be a jointly distributed random variables (discrete or continuous), then for any pair of positive integers (k 1, k 2 ) the joint moment of (X 1, X 2 ) of order (k 1, k 2 ) is defined to be: = x - x 1 2 ( ) ( ) k1 k2 x x p x x X X, if, are discrete k1 k2 x x f x x dx dx X X, if, are continuous

79 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 78 Let X 1 and X 2 be a jointly distributed random variables (discrete or continuous), then for any pair of positive integers (k1, k2) the joint central moment of (X 1, X 2 ) of order (k 1, k 2 ) is defined to be: k1 k2 ( µ ) ( µ ) ( ) x1 1 x2 2 p x1, x2 if X1, X 2 are discrete x1 x 2 = k1 k2 ( x1 µ 1) ( x2 µ 2 ) f ( x1, x2 ) dx1dx 2 if X1, X 2 are continuous - where

80 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Inequalities 7.1 Jensen inequality

81 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 80 Note that in general, Jensen inequality can be used to prove the Rao-Blackwell theorem The latter provides a method for improving the performance of an unbiased estimator of a parameter (i.e. reduce its variance cfr Chapter 5) provided that a sufficient statistic for this estimator is available. With g(x)=x 2 (hence g is a convex function), Jensen inequality says and therefore that the variance of X is always non-negative

82 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Markov s inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently) loose but still useful bounds for the cumulative distribution function of a random variable.

83 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 82 Markov's inequality gives an upper bound for the probability that X lies within the set indicated in red. Markov's inequality states that for any real-valued random variable X and any positive number a,, we have

84 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 83 Proof: Clearly, Therefore also Using linearity of expectations, the left side of this inequality is the same as Thus we have and since a > 0, we can divide both sides by a.

85 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Chebyshev s inequality

86 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 85

87 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Cantelli's inequality no exam material A one-tailed variant of Chebyshev s inequality with k > 0, is Proof: Without loss of generality, we assume. Let Thus for any t such that we have

88 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 87 The second inequality follows from Markov inequality: The above derivation holds for any t such that t+a>0. We can therefore select t to minimize the right-hand side:

89 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 88 An application: for probability distributions having an expected value and a median, the mean (i.e., the expected value) and the median can never differ from each other by more than one standard deviation. To express this in mathematical notation, let μ, m, and σ be respectively the mean, the median, and the standard deviation. Then (There is no need to rely on an assumption that the variance exists, i.e., is finite. This inequality is trivially true if the variance is infinite.)

90 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 89 Proof: Setting k = 1 in the statement for the one-sided inequality gives: By changing the sign of X and so of μ, we get Thus the median is within one standard deviation of the mean. Chebyshev inequality can also be used to prove the law of large numbers

91 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b Law of large numbers revisited

92 CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 91 Or formulated in the formal way

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