Question

Suppose the survival time X in weeks of a randomly selected male mouse exposed to 240 rads of gamma radiation has a gamma distribution with α = 8 and β = 15. What is the probability that a mouse survives between 60 and 120 weeks?

Answer 1

Answer:

1.3050 * 10 ^(-374)

Step-by-step explanation:

The pdf of a gamma distribution with parameters α = 8 and β = 15 is given by

$f(x;\alpha, \beta) =\frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)}$

Then , using the parameters, we have that, since $\Gamma(n) = (n-1)!$ for any integer n, then

$f(x) = \frac{15^8 x^{7} e^{-15x}}{7!}$

Then,

$\text{Pr}(60\leq X \leq 120 ) = \int_{60}^{120}\frac{15^8 x^7 e^{-15 x }}{7!} dx = 1.3050 * 10 ^{-374}$(the calculation of the antiderivative is performed by integration by parts, multiple times and it is beyond the scope of this answer).

Answer 2

Following are the solution to the given question:

Given:
$\alpha=8\\\\ \beta= 15\\\\ P(60To find:

probability=?

Solution:

[tex]\alpha=8 \\\\ \beta =15 \\\\ \to P(60Using the excel function that is "[tex]GAMMADIST(x, \alpha, \beta,true)$

         $=GAMMADIST(120,8,15, TRUE)-GAMMADIST(60,8,15,True) \\\\ =0.051134-0.547039 \\\\ =0.495906\\\\ =0.496$

So, the final answer is "0.496".

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