Answer:
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
