Question

Your driving time to work T (continuous random variable) is between 22 and 64 minutes if the day is sunny, and between 41 and 86 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 p subscript s = 0.41 and rainy with probability p subscript r equals 1 minus p subscript s. Your distance to work is X = 60 kilometers. Let V be your average speed for the drive to work, measured in kilometers per minute: V equals X over T Compute the value of the probability density function (PDF) of the average speed V at V = 0.62

Answer 1

Answer:

Given Data

Average driving time to work on a sunny day

The variance of average time to work on a sunny day

Average driving time to work on a rainy day

The variance of average time to work on a rainy day

Probability of a sunny day

Probability of a rainy day

Average driving time to work

T = 54.275min

The variance of driving time to work

The standard deviation of the time to work

For average speed to be equal to 0.62km/min. the time should be

The probability that the time will be equal to 96.774min is

P(T= 96.774)

Making the continuity correction

= P(96.274 < T< 97.274)

= P(3.7952 < z < 3.8855)

= P( z< 3.7952) - P(z < 3.8855)

= 0.000074 - 0.000051

= 0.000023

Hence, the probability that the average speed will be 0.62 km/min = 0.000023

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