Unformatted text preview: Chapter 4
Discrete Random Variables Two Types of Random Variables Random Variable: variable that assumes numerical values associated with random outcomes of an experiment Only one numerical value is assigned to each sample point. There are two types of random variables.
Discrete Continuous Two Types of Random Variables Discrete Random Variable: random variable that has a finite, or countable number of distinct possible values number of people born in a given month Continuous Random Variable: random variable that has an infinite number of distinct possible values height of trees in Winona 1. p ( x) > 0 for all values of x
2. p ( x) = 1 Probability Distributions for Discrete Random Variables Define an experiment as tossing 2 coins simultaneously. Let X represent the number of heads observed. X can assume values of 0, 1 or 2. Calculate the probability associated with each value of X. Probability Distributions for Discrete Random Variables Probability Distributions for Discrete Random Variables p( x = 0) = p( TT ) = 1 4 p( x = 1) = p( TH ) + p( HT ) = 1 4 + 1 4 = 1 2 p( x = 2) = p( HH ) = 1 4 Expected Values of Discrete Random Variables The mean, or expected value of a discrete random variable is: = ( x ) = xp( x )
Expected Value of X (number of heads observed)
X 0 1 2 Expected Value P(X) XP(X) 0 1 Expected Values of Discrete Random Variables The variance of a discrete random variable is: 2 2 2 = E x - ) = ( x - ) p( x) ( 1 2 1 2 1 2 1 = ( 0 - 1) + ( 1 - 1) + ( 2 - 1) = 4 2 4 2 The standard deviation is: = 2 =
1 2 Expected Values of Discrete Random Variables
Probability Rules for a Discrete Random Variable Probability Rules for a Discrete Random Variable
Chebyshev's Rule Applies to any distribution
p( - < x < + ) p( - 2 < x < + 2 ) p( - 3 < x < + 3 ) Empirical Rule Applies to mound shaped and symmetric distributions 0 .68 3 4 8 9 .95 1.00 The Binomial Random Variable Define a trial with two possible outcomes. Conduct n identical and independent trials. p(S) is constant for all trials. The Binomial Random Variable, X, equals the number of successes in n trials.
p = p(S) q = p(F) S = Success F = Failure The Binomial Random Variable The American Heart Association claims that only 10% of US adults over 30 can pass the President's Physical Fitness Commission's minimum requirements. Select 4 adults at random, administer the test. What is the probability that none of the adults passes the test? The Binomial Random Variable Use multiplicative rule to calculate probabilities of the possible outcomes p ( SSSS ) = ( 0.1) ( 0.1) ( 0.1) ( 0.1) = 0.0001 p ( FSSS ) = ( 0.9 ) ( 0.1) ( 0.1) ( 0.1) = 0.0009 ... p ( FFFF ) = ( 0.9 ) ( 0.9 ) ( 0.9 ) ( 0.9 ) = 0.6561 The Binomial Random Variable What is the probability that 3 of the 4 adults pass the test? p(3 of the 4 adults pass the test) = 4(.1)3(.9)=4(.09) = .0036 What is the probability that 3 of the 4 adults fail the test? p(3 of the 4 adults fail the test) = 4(.9)3(.1)=4(.0729) = .2916 The Binomial Random Variable Formula for the probability distribution P(x) n ( x ) = p x q n -x p x p = probability of success on single trial q = 1p n = number of trials x = number of successes in n trials The Binomial Random Variable Mean: = np = npq
2 Variance: Standard deviation: = npq ...
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- Fall '07
- Probability theory