Question

Your driving time to work  (continuous random variable) is between 27 and 69 minutes if the day is sunny, and between 47 and 87 minutes if the day is rainy, with a uniform probability density function in the given range in each case.

Assume that a day is sunny with probability  = 0.83 and rainy with probability .

Your distance to work is  = 50 kilometers. Let  be your average speed for the drive to work, measured in kilometers per minute:

Compute the value of the probability density function (PDF) of the average speed  at  = 0.69

Round your answer to five decimal digits after the decimal point.

Answer 1

Solution:-

Given that

Your driving time to work  (continuous random variable) is between 27 and 69 minutes if the day is sunny, and between 47 and 87 minutes if the day is rainy, with a uniform probability density function in the given range in each case.

Assume that a day is sunny with probability  = 0.83 and rainy with probability .

Your distance to work is  = 50 kilometers. Let  be your average speed for the drive to work, measured in kilometers per minute:

Compute the value of the probability density function (PDF) of the average speed  at  = 0.69

By the given condition

P(Sunny) =

P(rainy) =

So,

Also,

The pdf of driving time is

when X = 50 km and v = 0.69

Then

T = 72.46

Hence, the value of the probability density function of the average speed V at V = 0.69 is 0.00425

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