Chapter 3 Random Variables and Probability Distributions

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1 Math 322 Probabilit and Statistical Methods Chapter 3 Random Variables and Probabilit Distributions In statistics we deal with random variables- variables whose observed value is determined b chance. Random variables usuall fall into one of two categories: discrete or continuous Random Variable. A random variable (r.v.) is a function that associates a real number with each element in the sample space. Random variables will be denoted b uppercase letters and their observed numerical values b lowercase letters. Discrete Random Variable. A random variable is discrete if it can assume at most a finite or a countabl infinite number of possible values. Eample. Two balls are drawn in succession without replacement from an urn containing 4 red and 3 black balls. The possible outcomes and values of the random variable Y where Y is the number of red balls are: Sample Space RR 2 RB BR BB 0 Continuous Random Variable. A random variable is continuous if it can assume an value in some interval or intervals of real numbers and the probabilit that it assumes an specific value is 0. Discrete Probabilit Distributions Definition. The set of ordered pairs (, ( )) f is a probabilit function, probabilit mass function or probabilit distribution of the discrete random variable X if, () f ( ) 0 (2) f ( ) (3) P ( X ) f ( ). Eample. A committee of size 5 is to be selected at random from 3 chemists and 5 mathematicians. Find the probabilit distribution (p.d.) for the number of chemists on the committee.

2 Let X be the number of chemists on the committee. Then : 0,, 2, P ( X 0) f ( 0) ; P ( X ) f ( ) P ( X 2) f ( 2) P X Therefore the probabilit distribution of X is ; ( 3) f ( 3) f ( ) Eercise. Among 0 applicants for an open position 6 are female and are males. Suppose 3 applicants are randoml selected from the applicant pool for final interviews. Find the probabilit distribution for X, which is the number of female applicants among the final three. Eercise. Let w be a random variable giving the number of heads minus the number of tails in three tosses of a coin. (a) List the elements of the sample space (b) Assign a value w of W to each sample points. (c) Find the probabilit distribution of the random variable W assuming that the coin is biased so that a head is twice as likel to occur as a tail. Cumulative Distribution. The cumulative distribution F ( ) of a discrete random variable X with probabilit distribution f ( ) is ( ) ( ) ( ) F P X f t, for t. Eample 2. Find the cumulative distribution of the number of red balls in eample. Using 0 F ( ), show that f ( 3). F ; ( 0) f ( 0) 2

3 6 F ( ) f ( 0) + f ( ) 46 F ( 2) f ( 0) + f ( ) + f ( 2) ; F 3 f 0 + f + f 2 + f 3. ( ) ( ) ( ) ( ) ( ) Hence, 0, if < 0 if 0 < 6 F ( ) if < 2 46 if 2 < 3 if Now, f ( 3) F ( 3) F ( 2). Continuous Probabilit Distributions Definition. (Probabilit Densit Function) The function f ( ) is a probabilit densit function for the continuous random variable X, defined over the set of real numbers R, if () ( ) 0, f for all R (2) f ( ) d b (3) ( ) ( ) P a X b f d < <. Note that for a continuous random variable X, a a ( ) ( ) 0 P X a f d. a Cumulative Distribution. The cumulative distribution F ( ) of a continuous random variable X with densit function f ( ) is ( ) ( ) ( ) F P X f t dt for < <. 3

4 Eample. Suppose that a random variable X has a probabilit densit function given b k ( ) 0 f ( ) 0 elsewhere (a) Find the value of k that makes this a probabilit function. (b) Find P ( 0.4 X ) 0 ( ) (c) Find F ( ) P ( X ) k d k 6 ( 0.4) ( 0.4) 2 3 6( ) d and sketch the graph of this function. 0, if 0 F if, if 2 3 ( ) 3 2, 0 < < Eample 2. The weekl demand X for kerosene at a certain suppl station has a densit function given b for 0 f ( ) / 2 for < 2 0 elsewhere 3 (a) Find P X 2 2 F P X (b) Find ( ) ( ) Joint Probabilit Distributions Joint Probabilit Distributions of two discrete random variables X and Y : Definition. The function f (, ) variables X and Y if f, 0 for all (, () ( ) (2) ( ) f, (3) P ( X, Y f (, is a joint probabilit distribution of the discrete random For an region A in the plane, P ( X, Y ) A f (, A 4

5 Eample. Determine the value of c so that the following functions represent the joint probabilit distribution of the random variables X and Y. f, c for, 2,3,, 2,3 (a) ( ) (b) f (, c for 2, 0, 2, 2,3 Definition. The function f (, ) random variables X and Y if f, 0 for all (, () ( ) (2) f ( ), dd (3) (, ) (, ) P X Y A f dd for an region A in the A is a joint probabilit densit function of the continuous plane. Eample 2. Let X denote the reaction time, in seconds, to a stimulant and denote Y the temperature ( 0 F ) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint densit 4, 0 < < f (, 0, elsewhere f, is a probabilit densit function (a) Show that ( ) (b) Find P 0 X, Y Marginal Distributions Definition. The marginal distributions of X alone and Y alone are g f, and h f, for the discrete case and for the continuous case. ( ) ( ) ( ) ( ) ( ) (, ) and ( ) (, ) g f d h f d - - Eample. Suppose that X and Y have the following joint probabilit distribution: f (, ) 2 h( ) g ( ) 5

6 (a) Find the marginal distribution of X 2 g ( ) (b) Find the marginal distribution of Y 3 5 h( ) Statistical Independence Definition. Let X and Y be two random variables, discrete and continuous, with joint f, and marginal distributions g( ) and h( ) respectivel. The probabilit distribution ( ) random variables X and Y are said to be statisticall independent if and onl if for all (, ) within their range. f (, g( ) h( Eample. Let X denote the number of times a certain numerical control machine will malfunction:,2,3 times on an given da. Let Y denote the number of times a technician is called on an emergenc call. Their joint probabilit distribution is given as f(, ) (a) Find the marginal distribution of X The Marginal distribution of X is 2 3 g( ) (b) Find the marginal distribution of Y The marginal distribution of Y is 2 3 h( ) (c) Determine whether X and Y are independent or not. Let X and Y, then f (, Now check whether f (, g( ) h(.? So, f (,) g() h(), 0.05 (0.0)(0.20), which implies that X and Y are not independent. 6

7 Eample 2. Consider the following joint probabilit densit function of the random variables X and Y : 3, 3, 2 f (, < < < < 9 0 elsewhere (a) Find the marginal distributions of X and Y The marginal distribution of X is g( ) d , < < 3 6 The marginal distribution of Y is h( d 9 9 2, < < 2 9 (b) Are X and Y independent If f (, g( ) h( then X and Y are independent. Since , the variables X and Y are not independent. (c) Find P( X> 2) P( X 2) g( ) d d Eercise. A coin is tossed twice. Let Z denote the number of heads on the first toss and W the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 40% chance of occurring, find (a) the joint probabilit distribution of W and Z (b) the marginal distribution of W (c) the marginal distribution of Z (d) the probabilit that at least tail occurs. 7