Question

Refer to the continuous uniform distribution is given. Assume that a class length between 50.0 min and 52.0 min. is randomly selected, and find the probability that the given time. (a) Less than 51.5 min (b) exactly equal to 50.9

Answer 1

solution:  

Let X be the continuous uniform random variable representing the class length

Given that

50<= X <= 52

Let a = 50min , b = 52min

The probability density function (or) height of a function is given by

f(X) = 1/(b-a) = 1/(52-50) = 1/2

a)

Probability that X is less than 51.5 min = P(X<51.5)

= Area of region for 50<= X < 51.5

= Base * Height

= ( 51.5 - 50) * (1/2)

= 1.5 * (1/2)

= 0.75

$\therefore$  Probability that X is less than 51.5 min = 0.75

b)

Probability that X is exactly equal to 50.9 = P(X=50.9)

P(X=50.9) = 0 , since X is a continuous random variable when its interval decreases the probability of X is also decreases and approximates to 0 when the width of the interval becomes 0.

[observe, P(X=50.9) = Area of the region = width * height = 0 * 1/2 = 0 ]

$\therefore$ Probability that X is exactly equal to 50.9 = 0