Chapter 5. Random variables


 Jonathan Pearson
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1 Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like x to denote the various values that X can take discrete vs. continuous random variables a random variable is discrete if can only take on a countable number of distinct values, and continuous if it is characterized by an infinite range of values within some interval probability distribution function the function that assigns probabilities to events in which the random variable X takes on its possible values probability mass function the probability distribution function of a discrete random variable, which assigns a probability to each of the distinct values of the variable (we tabulate each value x along with the associated probability P (X = x)) 1
2 probability density function the probability distribution function of a continuous random variable, whose graph is a continuous curve that describes the likelihood that X takes on values that lie in various interval ranges cumulative distribution function the function that produces values of P (X x) for each possible value x of a (discrete or continuous) random variable X properties of a probability mass function Since the values P (X = x) of a probability mass function are probabilities, each must be a number between 0 and 1 The sum of all the values of a probability mass function must equal 1 2
3 expected value (E(X), or µ) for any discrete random variable X, the ideal (longrun) average value that X takes after observing infinitely many independent repetitions of X; computed from its probability mass function as the sum of the products of the values of X with their associated probabilities: E(X) = µ = x P (X = x) variance (V ar(x), or σ 2 ) for any discrete random variable X, the expected value of the squared deviations from µ of the values of X; computed from its probability mass function: V ar(x) = σ 2 = (x µ) 2 P (X = x) standard deviation (SD(X), or σ) for any discrete random variable X, the square root of its variance: SD(X) = σ = V ar(x) 3
4 Expectation and risk Uncertainty is viewed by consumers as risky; for instance, which of these three options would you go for: (1) a coin toss that determines which of two indistinguishable envelopes you are given, one of which contains $200 while the other requires you to pay a $100 penalty; (2) a coin toss that determines which of two indistinguishable envelopes you are given, one of which contains $100 while the other is empty; or (3) a single envelope which is known to contain $10? risk loving The risk loving consumer ignores risk and will seek the prospect with the highest possible reward, even if it threatens a negative expected gain (this person selects option #1 above) risk neutral The risk neutral consumer ignores risk and will accept any prospect that offers a positive expected gain (this person selects option #2 above) risk averse The risk averse consumer expects a reward for taking a risk (this person selects option #3 above) 4
5 Combining random variables and portfolio returns Investors build portfolios by distributing money over several investment options, but the return on each option can be viewed as a random variable (as its actual future return is unpredictable); assessing the return on the entire portfolio requires understanding the joint distribution of multiple random variables If X and Y are two random variables, and a and b are constants, then the variable ax + by, called a weighted combination of X and Y, has the following characteristics: its expected value is and its variance is E(aX + by ) = a E(X) + b E(Y ) V ar(ax+by ) = a 2 V ar(x)+2ab Cov(X, Y )+b 2 V ar(y ) 5
6 Thus, if a portfolio consists of investing a fraction w A of one s money in investment A (w A is also called the weight of investment A), and the remaining fraction w B in investment B, then the rate of return R p of the portfolio is directly related to the rates of return on the two investments, R A and R B : since R p = w A R A + w B R B, we have that the expected return on the portfolio is E(R p ) = w A E(R A ) + w B E(R B ), while the portfolio variance is V ar(r p ) = w 2 A V ar(r A )+2w A w B Cov(R A, R B )+w 2 B V ar(r B ) and the portfolio standard deviation is SD(R p ) = V ar(r p ) 6
7 Binomial random variables Bernoulli process series of independent and identical trials of an experiment which has only two outcomes, Success and Failure, and for which the probability p of Success (and therefore also the probability q = 1 p of Failure) is the same on each trial binomial random variable counts the number of Successes in a string of n trials of a Bernoulli process binomial probability mass function For x = 0, 1,..., n, we have ( ) n P (X = x) = p x q n x n! = x x!(n x)! px q n x binomial parameters if X is a binomial random variable, then E(X) = µ = np V ar(x) = σ 2 = npq SD(X) = σ = npq 7
8 Poisson random variables Poisson process the number of Successes of a series of independent and identical trials of an experiment take place during an interval of time or within a region of space so that the probability of Success is the same in all time intervals or spatial regions with equal duration or size Poisson random variable counts the number of Successes of a Poisson process in some time interval or spatial region Poisson probability mass function where µ measures the mean number of Successes of the Poisson process in the given time interval or spatial region, we have, for x = 0, 1,..., that P (X = x) = e µ µ x x! Poisson parameters if X is a Poisson random variable, then E(X) = µ V ar(x) = σ 2 = µ SD(X) = σ = µ 8
9 Hypergeometric random variables hypergeometric process a sample of n individuals is randomly selected without replacement from a population of size N containing exactly S Successes, in which n is a significant fraction of the size of N (so that distinct selections in the process are not independent of each other, and do not have the same probability of selecting a Success) hypergeometric random variable counts the number of Successes selected in a hypergeometric process hypergeometric probability mass function where a population of N individuals contain exactly S Sucesses, we have, for x = 0, 1,..., n, that ( S N S ) P (X = x) = x)( n x ( N n) 9
10 hypergeometric parameters if X is a hypergeometric random variable, then E(X) = µ = n S N V ar(x) = σ 2 = n S N SD(X) = σ = n S N ( 1 S ) N n N N 1 ( 1 S N ) N n N 1 10
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