Question

Between workouts recovery, the end of one and start of other is 90 sec on average. Assume the recovery times are independent and identically distributed.

a. A recovery is short if it lasts less than 30 secs. Find the probability that the recovery time is short. Explain choice of random variable to model the problem

B. Suppose there are 30 recovery’s during one workout (assume number of recoveries are independent of recovery time for each). Find the probability fewer than 5 total recoveries had a short recovery time. Use a.

Answer 1

Let X be the random variable denoting the recovery time (in seconds).

Thus, X ~ Uniform (0,90)

The PDF of X is, f(x) = , 0 < x < 90 and = 0, otherwise.

a. The probability that a recovery time is short = P(X < 30) = = = 0.3333 (Ans).

b. Let Y be the random variable denoting the number of short recovery time out of 30 recoveries.

Thus, X ~ Bin(30, 0.3333)

The probability that there are fewer than 5 short recoveries = P(X < 5) = P(X 4) = = 0.0122 (Ans).