The amount of time it takes for a student to complete a statistics quiz is uniformly distributed (or, given by a random variable that is uniformly distributed) between 32 and 64 minutes. One student is selected at random. Find the probability of the following events.
A. The student requires more than 59 minutes to complete the quiz.
Probability =
B. The student completes the quiz in a time between 37 and 43 minutes.
Probability =
C. The student completes the quiz in exactly 44.74 minutes.
Probability =

a)  P( t > 59 ) = 0.15625

b) P ( 37 < t < 43 ) =0.1875

c) P ( t = 44.74 ) = 0.03125

Step-by-step explanation:

Given:

- The amount of time taken to give the quiz has a random variable X that follows a uniform distribution over the interval [ 32 , 64 ] minutes.

Find:

Find the probability of the following events.

A. The student requires more than 59 minutes to complete the quiz.

B. The student completes the quiz in a time between 37 and 43 minutes.

C. The student completes the quiz in exactly 44.74 minutes.

Solution:

- The probability mass function of a uniform distribution or interval [ a , b ] is given by:

                              pmf f(x) = 1 / ( b - a )

                              f(t) = 1 / ( 64 - 32 )

                              f(t) = 1 / 32

-  The student requires more than 59 minutes to complete the quiz f ( t > 59):

                              CDF F(x) = ( x - a ) / ( b - a )

                              CDF F(t) = ( t - 32 ) / 32

                              P( t > 59 ) = F(64) - F(59)

                              = ( 64 - 32 ) / 32 - ( 59 - 32 ) / 32

                              = 1 - 27/32

                               = 0.15625

- The student completes the quiz in a time between 37 and 43 minutes.

                              P ( 37 < t < 43 ) = F(43) - F(37)

                              = ( 43 - 32 ) / 32 - ( 37 - 32 ) / 32

                              = 3/16

                              = 0.1875

- The student completes the quiz in exactly 44.74 minutes.

The probability for each minute is uniform or constant for the distribution given by pmf:

                               P ( t = 44.74 ) = f(44.74)

                               = 1 / 32

                               = 0.03125

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