Question

The time it takes for you to run around a pond near your house is uniformly distributed between 30 and 40 minutes on sunny days, and uniformly distributed between 40 and 60 minutes on rainy days. Assume that a day is sunny with probability 2/3 and rainy with probability 1/3. Find the mean, and the variance of the time it takes for your run.

Answer 1

Let X denote the time taken to run on sunny days

and Y denote the time taken to run on rainy days

Mean of X = (30 + 40)/2 = 35 minutes

Variance of X = = 25/3 = 8.33

Mean of Y = (40 + 60)/2 = 50 minutes

Variance of Y = = 100/3 = 33.33

The time to run can be expressed as (2/3)X + (1/3)Y

Thus, Mean time = 2/3*E(X) + 1/3*E(Y)

= 2/3*35 +1/3*50

= 40 minutes

Variance of the time taken = 4/9*Var(X) + 1/9*Var(Y)

= 4/9*25/3 + 1/9*100/3

= 200/27 = 7.41