Question

Your driving time to work T (continuous random variable) is between 24 and 66 minutes if the day is sunny, and between 49 and 82 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 Ps = 0.64 and rainy with probability Pr = 1 -Ps.

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

V=T/X

Compute the value of the probability density function (PDF) of the average speed V at V = 0.67

Answer 1

Answer:

Given that,

Your driving time to work T (continuous random variable) is between 24 and 66 minutes if the day is sunny, and between 49 and 82 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 Ps = 0.64 and rainy with probability Pr = 1 -Ps.

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

V=T/X

Compute the value of the probability density function (PDF) of the average speed V at V = 0.67:

The probability density function of T is,

for and f(t)=0 otherwise V=50/T

f(t)=(0.64/42)+ (0.36/33)

The probability density function of V is,

Where t=w(v) and and t=w(v)=50/v

Therefore,

for and f(v)=0 otherwise.

For v=0.67:

f(v)=(0.0152+0.0109)(50/(0.67)^2)

=0.0261 111.3834

f(v) =2.9071 (Approximately)