Question

Practice B (all answers to 4 decimal places)

a) Let X and Y be the lifetimes of the two components of a standby system, and suppose X and Y are independent exponentially distributed random variables with expected lifetimes 3 weeks and 4 weeks, respectively. The combined lifetimes of the two components equal the total lifetime of the system. Let T=X+Y, the lifetime of the standby system. What is the standard deviation of the lifetime of the system?

b) Let U1,U2,...,Un be independent and identically distributed (i.i.d.) Uniform(a=0,b=1) random variables. Given n = 300 and N = U1+ ...+ Un, Calculate the expected value and standard deviation, E[U1+...+Un] and SD[U1+...+Un]] in terms of n.

Answer 1

a)

We are given that X and Y are exponentially distributed random variable which are independent and have expected value 3 weeks and 4 weeks respectively. Thus, we get:

X ~ exponential(λ = 1/3) and Y ~ exponential(μ = 1/4) where λ and μ are rate parameters of the distribution of X and Y respectively.

Now, we are given that the lifetime of the standby system is given by: T = X+Y. Thus, the standard deviation of the lifetime of the standby system is given by:

b)

We are given that U1,U2,...,Un are independent and identically distributed (i.i.d.) Uniform(a=0,b=1) random variables. Thus, the mean and variance of Ui's is given by:

Now, the required expected value is given by:

Now, the required standard deviation is given by:

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