Question

You are given the following information: If X ∼ U[0, 1], E[X] = 1/2 and σ^2 (X) = 1/12 . Let Y ∼ U[100, 700]. Find E[Y ] and σ^2 (Y ) as easily as possible, using the information given.

Would someone give the full and correct answer to this problem please?

Answer 1

According to the question, Y follows Uniform distribution with parameters 100 & 700. So, the probability density function of the random variable Y is given by-

100y700 600 f(Y)700-100

  

0, otherwise

We are to compute E(Y) &   .

-o2(Y)

T00 dy 100 600 E(Y) =

Y00 1 ydy 600 J100

1 y2 700 100 2 600

:(7002 1002) 1200

= 400

Now, i.e. variance of Y is given by -

-o2(Y)

-(E(Y) Var (Y)= E(Y)

r700 2 E(Y2) -dy 100 600

$= \frac{1}{600}\int_{100}^{700}\ y^{2}dy$

$= \frac{1}{600}[\frac{y^{3}}{3}]_{100}^{700}$

$= \frac{1}{1800}[700^{3}- 100^{3}]$

Also,  

Consequently,

Var(Y) 190000 160000 30000

Therefore,  E[Y ] and σ^2 (Y ) will have the values 400 & 30000 respectively.