Question

(1 point) Suppose a random variable ?x is best described by a uniform probability distribution with range 11 to 44. Find the value of ?a that makes the following probability statements true.

(a)  ?(?≤?)=0.26
?=

(b)  ?(?<?)=0.95
?=

(c)  ?(?≥?)=0.65
?=

(d)  ?(?>?)=0.25
?=

(e)  ?(1.49≤?≤?)=0.06

a=

Answer 1

Solution:

We are given that random variable follows Uniform distribution.

A = 11

B = 44

Range = B – A = 44 – 11 = 33

P(x<a) = (x – a)/(b – a)

P(x>a) = (b – x)/(b – a)

Part a

We are given

P(x≤a) = 0.26

(a – A)/(B – A) = 0.26

(a – 11)/(44 – 11) = 0.26

(a – 11)/33 = 0.26

(a – 11) = 33*0.26

(a – 11) = 8.58

a = 8.58 + 11

a = 19.58

Part b

P(x<a) = 0.95

a = (B – A)*0.95 + A

a = 33*0.95 + 11

a = 42.35

Part c

P(x≥a) = 0.65

(B – a)/(B – A) = 0.65

(44 – a) = (B – A)*65

-a = 33*0.65 - 44

-a = -22.55

a = 22.55

Part d

P(x>a) = 0.25

(44 – a) = (B – A)*0.25

-a = 33*0.25 - 44

a = 35.75

Part e

?(1.49≤?≤?)=0.06

Range is from 11 to 44, so we have

?(11≤?≤?)=0.06

(a – 11) = (B – A)*0.06

(a – 11) = 33*0.06

a = 11 + 33*0.06

a = 12.98