Alvin’s driving speed to work is uniform between 40 and 45 miles/hour if the day

Question

Alvin’s driving speed to work is uniform between 40 and 45 miles/hour if the day is sunny, and uniform between 30 and 40 miles/hour if the day is rainy. Assume that a day is sunny with probability 2/3 and rainy with probability 1/3.  

(a) What is the PDF of the driving speed, viewed as a continuous random variable X?  

(b) What is Alvin’s average driving speed to work?  

(c) Alvin lives 10 miles away from work, what is Alvin’s average driving time to work? Hint: the PDF of X is piecewise constant in the two intervals.

Explanation / Answer

Let X follows Uniform Distribution [20,30]

f(x) =(1 / (30-20) =1/10

b) Mean =(a+b) /2 =(20+30)/2 =25

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