Question

The random variable X milliseconds is the total access time (waiting time + reading time) to obtain a block of information from a computer disk. X is evenly distributed between 0 and 12 milliseconds. Before making a determined task, the computer must access 12 different information blocks from the disk. (The access times for different blocks are independent one of the other.) The total access time for all information is a random variable A milliseconds.

(1) What is the E [X], the expected value of the access time?

(2) What is the Var [X], the variance of the access time?

(3) What is the E [A], the expected value of the total access time?

(4) What is the σA, the standard deviation of the total access time?

(5) Use the central limit theorem to estimate P [A> 75 ms], the probability that the Total access time exceeds 75 ms.

(6) Use the central limit theorem to estimate P [A <48 ms], the probability that the Total access time is less than 48 ms

Answer 1

Since X is evenly distributed between 0 and 12 milliseconds so

1)   Using mean formula of uniform distribution

2) Using variance formula for uniform distribution.

3) As A is the total access time for all information that means A is the sum of the 12 blocks.

Hence

4) . So

5) Define the standard random variable Z as

.

Then using normal table we have

6)