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 40 weeks and variance 320 weeks.

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?

The Gamma distribution "is a continuous, positive-only, unimodal distribution that encodes the time required for [tex]\alpha[/tex] events to occur in a Poisson process with mean arrival time of [tex]\beta[/tex]"

Solution to the problem

Let X the random variable that represent the lifetime for transistors

For this case we have the mean and the variance given. And we have defined the mean and variance like this:

[tex]\mu = 40 = \alpha \beta[/tex] (1)

[tex]\sigma^2 =320= \alpha \beta^2[/tex] (2)

From this we can solve [tex]\alpha[/tex] and [/tex]\beta[/tex]

From the condition (1) we can solve for [tex]\alpha[/tex] and we got:

[tex]\alpha= \frac{40}{\beta}[/tex] (3)

And if we replace condition (3) into (2) we got:

[tex]320= \frac{40}{\beta} \beta^2 = 40 \beta[/tex]

And solving for [tex]\beta = 8[/tex]

And now we can use condition (3) to find [tex]\alpha[/tex]

[tex]\alpha=\frac{40}{8}=5[/tex]

So then we have the parameters for the Gamma distribution. On this case [tex]X \sim Gamma (\alpha= 5, \beta=8)[/tex]

Part a

For this case we want this probability:

[tex]P(1 \leq X \leq 40)[/tex]

In order to find this probability we can use excel with the following code:

=GAMMA.DIST(40;5,8,TRUE)-GAMMA.DIST(1,5,8,TRUE)

And we got:

[tex]P(1 \leq X \leq 40)=0.560[/tex]

Part b

For this case we want this probability:

[tex]P(X \geq 40)=1-P(X<40)[/tex]

In order to find this probability we can use excel with the following code:

=1-GAMMA.DIST(40,5,8,TRUE)

And we got:

[tex]P(X \geq 40)=1-P(X<40)=0.440[/tex]