Question

11. Let X1, X2, X3 and X4 be independent lifetimes of memory chips. Suppose that Xi N(300, 102) for i = 1, 2, 3, 4 where the parameters are measured in hours. Compute the prob- ability that at least two of the four chips lasts at least 310 hours. (You may leave your answer in terms of an integral, in terms of, or you may leave your answer as an actual real number).

Answer 1

For each of the 4 components, the lifetime distribution here is given as:

$X_i \sim N(\mu = 300, \sigma = 10)$

Therefore the probability that any of the components lasts at least 310 hours is computed here as:

Converting it to a standard normal variable, we have here:

$P(Z > \frac{310 - 300}{10})$

P(Z > 1)

This is obtained from the standard normal tables as:

$P(Z > 1) = 1 - \Phi (1)$

Now the probability that out of 4 such components, at least 2 of the four chips lasts at least 310 hours is computed using Binomial probability function as:

$Y \sim Bin(n = 4, p= 1 - \Phi(1))$

$P(Y \geq 2) = 1 - P(Y = 0) - P(Y = 1)$

$P(Y \geq 2) = 1 - [\Phi(1)]^4 - 4*[\Phi(1)]^3(1 - \Phi(1))$

This is the required probability expression here.