Suppose that when a transistor of a certain type is subjected to an accelerated life test, the lifetime X (in weeks) has a gamma distribution with mean of 40 and variance of 320.
a. What is the probability that a transistor will last between 1 and 40 weeks?
b. What is the probability that a transistor will last at most 40 weeks? Is the median of the lifetime distribution less than 40? Why or why not?

Question

Quizplus · Accepted Answer

The Answer is

\mu = 40 , \sigma ^ { 2 } = 320 \Rightarrow \alpha \beta = 40 , \alpha \beta ^ { 2 } = 320 \Rightarrow \beta = 8 , \alpha = 5

a.

P ( 16 \leq X \leq 40 ) = F ( 5 ; 5 ) - F ( 2 ; 5 ) = .507

b.

P ( X \leq 40 ) = F ( 5 ; 5 ) = .560

so while the mean is 40, the median is less than 40.

( P ( X \leq \bar { \mu } ) = 5 )

This is the result of the positive skew of the gamma distribution.

Suppose That When a Transistor of a Certain Type Is