STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 2 1 2 2 2 3 A random variable associates a numerical value with each outcome of a random experiment A random variable is said to be discrete if it has either a finite number of values or infinitely many values that can be arranged in a sequence If a random variable represents some measurement on a continuous scale and therefore capable of assuming all values in an interval it is called a continuous random variable The probability distribution of a discrete random variable is a list of all its distinct numerical values along with their associated probabilities x x1 x2 x3 f x f x1 f x2 f x3 xn f xn 1 for each x 0 f x 1 2 f x 1 all x 1 00 Often a formula can be used in place of a detailed list 1 A balanced fair coin is tossed twice Let X denote the number of H s Construct the probability distribution of X x f x 2 a Suppose a random variable X has the following probability distribution x f x 10 0 20 11 0 40 12 0 30 13 0 10 Find the expected value of X E X x f x 10 0 2 11 0 4 12 0 3 13 0 1 E X X x f x x f x all x b Find the variance of X Var X x f x 10 0 2 11 0 4 12 0 3 13 0 1 Var X X2 x X 2 f x all x E X X 2 x X x X 2 f x x f x 10 0 2 11 0 4 12 0 3 13 0 1 Var X X2 x 2 f x x2 f x E X 2 all x E X 2 E X 2 c Find the standard deviation of X SD X SD X X d 2X Find the cumulative distribution function of X F x P X x x f x 10 0 2 11 0 4 12 0 3 13 0 1 F x E g X g x f x E a X b a E X b all x Var a X b a Var X 2 SD a X b a SD X 3 Suppose E X 7 SD X 3 a Y 2 X 3 Find E Y and SD Y b W 5 2 X Find E W and SD W 4 Suppose a discrete random variable X has the following probability distribution P X 0 2 e a Find E X b Find Var X P X k 1 k 1 2 3 k 2 k
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