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a. Between 5 and 10 minutes b. Between 8 and 20 minutes c. Between 1 and 45 minutes d. More than 60 minutes Math 135 Class Notes – Week 12 Bittenger 10th Ed. 5 5.5 Probability: Expected Value; The Normal Distribution () OBJECTIVE: Find E ( x ) , E x 2 , then mean, the variance, and the standard deviation.
Evaluate normal distribution probabilities using a table. Calculate percentiles for a normal
distribution. Expected Value DEFINITION: Let x be a continuous random variable over
the interval [ a, b ] with probability density function f. The
expected value of x is defined by
E ( x ) = ∫ x ⋅ f ( x ) dx .
b a DEFINITION: The expected value of g ( x ) is defined by
E ( g ( x )) = ∫ g ( x ) ⋅ f ( x ) dx
b a where f is a probability density function for x.
Example 1: Given the probability density function f ( x ) = 1 − 1 x , over [ 0, 4 ] , find
8 E ( x ) and E ( x 2 ) . Math 135 Class Notes – Week 12 Bittenger 10th Ed. 6 DEFINITION: The mean, µ , of a continuous random variable x, is
defined to be E ( x ) . That is, µ = E ( x ) = ∫ xf ( x ) dx
b a where f is a probability d...
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