Question

a machine has n identical components each of which has an exponentially distributed lifetime x with expected value 10. the total life of the n components is also a random variable, since it is a function of random variables. we denote that total life as t.

Answer 1

Explanation:

That is correct. The total life of the n identical components is a random variable and can be expressed as the sum of the lifetimes of each component, that is:

t = x1 + x2 + ... + xn

where x1, x2, ..., xn are identically and independently distributed exponential random variables with the same mean of 10.

The expected value of each component's lifetime is given by E(x) = 1/λ, where λ is the rate parameter of the exponential distribution. Since the expected value of x is 10, we have:

E(x) = 1/λ = 10

=> λ = 1/10

Thus, each component's lifetime follows an exponential distribution with rate parameter λ = 1/10.

The expected value of the total life t is then given by:

E(t) = E(x1 + x2 + ... + xn)

= E(x1) + E(x2) + ... + E(xn) (by linearity of expectation)

= n * E(x)

= n * (1/λ)

= 10n

Therefore, the expected value of the total life of the n identical components is 10n.