Question

A continuous random variable is uniformly distributed between 50 and 75.

a. What is the probability a randomly selected value will be greater than 65​?

b. What is the probability a randomly selected value will be less than 60​?

c. What is the probability a randomly selected value will be between 60 and 65​?

a.

​P(x>65​)=

​(Simplify your​ answer.)

b.

​P(x<60​)=

​(Simplify your​ answer.)

c.

​P(60<x<65​)=

​(Simplify your​ answer.)

Answer 1

X is a continuous random variable that is uniformly distributed between 50 and 75

PDF

f(x) = 1/(75 - 50) , 50 < x < 75

= 0 , otherwise   

f(x) = 1/25 ,  50 < x < 75

= 0 , otherwise

f(x) = 0.04 , 50 < x < 75

= 0 , otherwise

Distribution function

DF : P(X $\leq$ x ) = F(x) = x - 50/(75 - 50)  

P(X $\leq$ x ) = (x - 50)/25

a)

P(X > 65) = 1 - P(X $\leq$ 65)

= 1 - (65 - 50)/25

= 1 - 0.6

= 0.4

b)

P(X < 60) = (60 - 50)/25

= 10/25

= 0.4

c)

P(60 < X < 65) = P(X < 65) - P(X < 60)

= (65 - 50)/25 - (60 - 50)/25

= 15/25 - 10/25

= 0.6 - 0.4

= 0.2

NOTE

If x is uniformly distributed between a and b then

PDF : f(x) = 1/(b - a) , a< x < b

= 0 , otherwise

DF : F(x) = P(X$\leq$ x) =  $\int_{a}^{x}1/(b- a)dt$

= t/(b - a)$\mid _{a}^{x}$

= 1/(b-a)[x - a]

= (x - a)(b -a)