Discrete and Continuous Random Variables. Summer 2003

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1 Discrete and Continuous Random Variables Summer 003

2 Random Variables A random variable is a rule that assigns a numerical value to each possible outcome of a probabilistic experiment. We denote a random variable by a capital letter (such as X ) Examples of random variables: r.v. X: the age of a randomly selected student here today. r.v. Y: the number of planes completed in the past week Summer 003

3 Discrete or Continuous A discrete r.v. can take only distinct, separate values Examples? A continuous r.v. can take any value in some interval (low,high) Examples? Summer 003 3

4 Discrete Random Variables A probability distribution for a discrete r.v. X consists of: Possible values x, x,..., x n Corresponding probabilities p, p,..., p n with the interpretation that p(x = x ) = p, p(x = x ) = p,..., p(x = x n ) = p n Note the following: Variable names are capital letters (e.g., X) Values of variables are lower case letters (e.g., x ) Each p i 0 and p + p p n = Summer 003 4

5 Probability tree and probability distribution for r.v. X (total # Heads in experiment ) Outcome X (#Hs) Probab H3 HHH H H P(X = 0) = /8 P(X = ) = 3/8 P(X = ) = 3/8 P(X = 3) = /8 T3 H3 T T3 0.5 H3 H T 0.5 T3 HHT3 0.5 HTH3 0.5 HTT3 0.5 THH3 0.5 THT H3 T 0.5 TTH T3 TTT Summer 003 5

6 A histogram is a display of probabilities as a bar chart r.v. X = number of heads when tossing 3 unbiased coins 5/8 4/8 3/8 /8 / x Now let s consider experiment : number of heads when tossing 3 biased coins (p(h) = 0.30). Again r.v. X: total number of heads obtained when performing experiment Summer 003 6

7 Probability tree and probability distribution for r.v. X (total # Heads in experiment ) H3 H T3 H 0.7 Outcome X (#Hs) Probabil HHH HHT H3 T 0.3 HTH X p(x) T HTT H3 THH H T 0.7 T3 THT H3 T 0.3 TTH T3 TTT Summer 003 7

8 Histogram for experiment r.v. X = total number of heads when tossing 3 biased coins with p(h) = /8 p(x) 4/8 3/8 /8 /8 0/ x x p(x) Summer 003 8

9 Example : Let Let X be the random variable that denotes the number of orders for aircraft for next year. Suppose that the number of orders for aircraft for next year is estimated to obey the following distribution: Orders for aircraft next year x i Probability p i Summer 003 9

10 A histogram is a display of probabilities as a bar chart A histogram is a display of probabilities as a bar chart 0.35 Probability Distribution of the Number of Orders of Aircraft Next Year 0.30 Probability Number of Orders Summer 003 0

11 Eastern Division Sales ($ million) Probability Western Division Sales ($ million) Probability X and Y denote the sales next year in the eastern division and the western division of a company, respectively. X and Y obey the following distributions: Probability Probability Probability Distribution Function of Eastern Division Sales Sales ($ million) Probability Distribution Function of Western Division Sales Sales ($ million) Summer 003

12 Consider two random variables X, Y, with the following histograms: (A) (B) P(X =x) P(Y = y) x y Random variable X Random variable Y How do we describe and compare X and Y? Summer 003

13 Summary Statistics for a r.v.: Three important measures - Mean or Expected Value: Represents average outcome; a measure of central tendency - Variance: E(X) = µ X = n i= P(X = x )x Squared deviation around the mean; a measure of spread i i = n i= p x i i Var(X) - Standard Deviation : n n = σ X = P(X = xi)(xi µ X ) = i= i= p (x Square root of the variance. A measure of spread in the same units as the random variable X. i i µ X ) SD(X) = σ X = σ X Summer 003 3

14 Example: Let X be the number that comes up on a roll of one die. Compute the mean, variance and standard deviation of X. Outcome P(X=x) /6 /6 3 /6 4 /6 5 /6 6 /6 µ x = /6* +/6 * +/6 *3 + /6*4+/6*5+/6*6 = 3.5 (σ x ) = /6*(-3.5) +/6 *(-3.5) +/6 *(3-3.5) + /6 *(4-3.5) +/6 *(4-3.5) +/6 *(5-3.5) +/6 *(6-3.5) =.97 σ x = Sqrt{.97}= Summer 003 4

15 Example : Compute the mean, variance,standard deviation of X (eastern division sales) and Y (western division sales) µ X = $5. mil µ Y = $5.5 mil σ X =.06 mil σ Y =.3875 mil σ X = $.09 mil σ Y = $. 545mil Summer 003 5

16 Continuous Random Variables A continuous random variable can take any value in some interval Example: X = time a customer spends waiting in line at the store Infinite number of possible values for the random variable. How can we describe a probability distribution? (We can no longer list the p i s and x i s!) For a continuous random variable, questions are phrased in terms of a range of values. Example: We might talk about the event that a customer waits between 5.0 and 0.0 minutes, and not about the event that a customer waits exactly 5.5 minutes! (why?) Summer 003 6

17 Continuous Probability Distributions Continuous R.V. s have continuous probability distributions known also as the probability density function (PDF) Since a continuous R.V. X can take an infinite number of values on an interval, the probability that a continuous R.V. X takes any single given value is zero: : P(X=c)=0 Probabilities for a continuous RV X are calculated for a range of values: P (a X b) P (a X b) is the area under the probability distribution function, f(x), for continuous R.V. X. The total area under f(x) is.0 a b c Summer 003 7

18 The Uniform Distribution X is uniform on [a,b] if X is equally likely to take any value in the range from a to b. Example: Suppose that transit time of the subway between Alewife Station and Downtown Crossing is uniformly distributed between 0.0 and 0.0 minutes. a) What is the mean transit time? E(X) = µ x =? b) What is the probability that the transit time exceeds.0 minutes? P(X >=.0) =? c) What is the probability that the transit time is between 4.0 and 8.0 minutes? P(4<= X <= 8) =? Summer 003 8

19 Uniform Distribution A continuous R.V. X is uniform in the interval [a-b] if it is equally likely to take any value in the interval [a,b]. Thus f(x) for a continuous uniform R.V. is a horizontal line (i.e., a constant). f(x)=/(b-a) so that the area under the curve is.0. f ( x) = b a for a x b 0 for all other values f ( x) Area = b a a x b Summer 003 9

20 Uniform Distribution transit time of the subway between Alewife Station and Downtown Crossing is uniformly distributed between 0.0 and 0.0 minutes. f ( x) = for for 0 x 0 all other valu es f ( x) b 0 a P(X >=.0) = 0.80 P(4<= X <= 8) = Area = x Summer 003 0

21 Lot Weights in a Warehouse are Uniformly distributed Between 4 and 47 lbs. f ( x) = for 4 x for all other values f ( x) 47 4 = 6 Area = 4 47 x Summer 003

22 What is the probability that the weight of a randomly selected lot is between 4 and 45 lbs? ( x X x ) = b a ( x x ) P P( 4 X 45) = ( 45 4) = 6 f ( x) / 6 45 = 6 ( 4 ) Area = x Summer 003

23 Uniform Distribution Mean and Standard Deviation Mean Mean µ = a + b µ = = = 44 Standard Deviation σ = b a Standard Deviation σ = = = Summer 003 3

24 5.063 Summer Summer Cumulative Distribution Function (CDF) ) ( ) ( ) ( t F dx x f t X P t = = t F(t) t ) ( ) ( s X P s F = ( s) X P F(s) s s ) ( ) ( ) ( ) ( ) ( s F t F s X P t X P t X s P = < =

25 Let X be uniformly distributed on [a,b]. Then density function of X is: f(x) f ( x) = for a x b b a 0 for all other values /(b-a) Find the CDF: a b F ( x) = ( b a) x a ( b a) 0 x F ( x ) = f ( u ) du for for a x x < x > a b b F(x) a b Summer 003 5

26 Summary Discrete and continuous random variables have to be understood differently Histograms are very useful for discrete variables Next session we will talk about binomial distributions Uniform continuous random variables are a good place to start for our later work with normal distributions etc Summer 003 6

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