Question

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 =

Answer 1

(A) 0.15625

(B) 0.1875

(C) Can't be computed

Explanation:

We are given that the amount of time it takes for a student to complete a statistics quiz is uniformly distributed between 32 and 64 minutes.

Let X = Amount of time taken by student to complete a statistics quiz

So, X ~ U(32 , 64)

The PDF of uniform distribution is given by;

f(X) =
`(1)/(b-a)` , a < X < b where a = 32 and b = 64

The CDF of Uniform distribution is P(X <= x) =
`(x-a)/(b-a)`

(A) Probability that student requires more than 59 minutes to complete the quiz = P(X > 59)

P(X > 59) = 1 - P(X <= 59) = 1 -
`(x-a)/(b-a)` = 1 -
`(59-32)/(64-32)` =
`1-(27)/(32)` = 0.15625

(B) Probability that student completes the quiz in a time between 37 and 43 minutes = P(37 <= X <= 43) = P(X <= 43) - P(X < 37)

P(X <= 43) =
`(43-32)/(64-32)` =
`(11)/(32)` = 0.34375

P(X < 37) =
`(37-32)/(64-32)` =
`(5)/(32)` = 0.15625

P(37 <= X <= 43) = 0.34375 - 0.15625 = 0.1875

(C) Probability that student complete the quiz in exactly 44.74 minutes

= P(X = 44.74)

The above probability can't be computed because this is a continuous distribution and it can't give point wise probability.