Question

Suppose the lifetime, in years, of a motherboard is modeled by a Gamma distribution with parameters α=80 and λ=4. Use the Central Limit Theorem to approximate the probability that the motherboard of a new computer will last for at least the next 15 years.

Answer 1

Answer:

Therefore there's a 99.99% probability the motherboard of your new computer will last for at least 15 years.

Step-by-step explanation:

This is the general idea to solve the problem.

Suppose that the mean and variance of the your distribution are .$\mu , \sigma$ respectively. Then, according to the problem you are looking for the probability.

$P(X \geq 15) = 1 - P(X$

Consider then the following random variable.

$T = X_1 + X_2 + .... + X_{14}$

Using the central limit theorem  $T$ distribution will be close to normal, and its mean and variance will be  $14\mu , 14\sigma$  , respectively. Therefore you just have to find the probability that a normally distributed  random variable with that mean and that variance which I just mentioned is less than 14.

For this case we have that

$\mu = \alpha / \gamma = 20 \\\sigma = \alpha(\alpha+1) / \gamma^2 - (\alpha/ \gamma)^2 = 5$

Then you have that

$14\mu = 280\\14\sigma = 70$

and we have that if  $N$ is a normally distributed random variables with mean 280 and variance 70 we have that

$P(N \leq 14 ) = 0.0001$

the actual probability we are looking for is

$1-P(N\leq14) = 1-(0.0001) = 0.9999$

Therefore there's a 99.99% probability the motherboard of your new computer will last for at least 15 years.