Question

Let U be a uniform continuous random variable on the interval [2, 8].

(a) What is P(U = 4)?

(b) What is P(U ≤ 4)?

(c) What is P(4 ≤ U ≤ 7)?

(d) Find a formula for FU(x).

(e) Find a formula for fU(x).

(f) What is E(U)?

(g) What is Var(U)?

(h) What is E(1 − U 2 )?

Answer 1

Solution:

We are given that the random variable U follows uniform distribution with a = 2 and b = 8.

Part a

P(U=4) = 0.00

...because exact probability for any continuous distribution is always 0.00.

Part b

We have formula as below:

P(U≤x) = P(U<x) = (x – a)/(b – a)

So, by using above formula,

P(U≤4) = (4 – 2)/(8 – 2) = 2/6 = 0.333333

Required probability = 0.333333

Part c

We have formula as below:

P(m≤U≤n) = P(m<U<n) = (n – m) / (b – a)

So, by using above formula,

P(4≤U≤7) = (7 – 4)/(8 – 2) =3/6 = ½ = 0.50

Required probability = 0.5000

Part d

Here, we have to find CDF FU(x)

CDF FU(x) = (x – a) / (b – a)

FU(x) = (x – 2)/(8 – 2)

FU(x) = (x – 2)/6

Part e

Here, we have to find PDF fU(x)

PDF fU(x) = 1/(b – a)

fU(x) = 1/(8 – 2)

fU(x) = 1/6

Part f

E(U) = (a + b)/2 = (2 + 8) / 2 = 10/2 = 5

E(U) = 5

Part g

Var(U) = (b – a)^2/12

Var(U) = (8 – 2)^2/ 12

Var(U) = 6^2/12

Var(U) = 36/12

Var(U) = 3