The length of time in microseconds for a computer to perform a task of type i has a geometric distribution (1 − pi)p n−1 i , n ≥ 1, with mean 1/(1−pi), for i = 1,...,. Also, the type of task to be...

The length of time in microseconds for a computer to perform a task of type i has a geometric distribution (1 − pi)pⁿ⁻¹
i , n ≥ 1, with mean 1/(1−pi), for i = 1,...,. Also, the type of task to be worked on is a random variable X, where p(i) = P{X = i} is known for i = 1,...,m. Under these assumptions, the time T to perform a task has the conditional probability measure P{T = n|X = i} = (1 − pi)pⁿ−1 i . Find expressions for E[T |X = i], P{T = n} and ET .