Question

2- Choose the correct aaswer (Write dowu the correct answer letter AND valuc at your answer a. | 1f random variable Xhas a mea Mr O a standard deviation ơx 4, and (C) 2 and σ r 16 (1) μ,--3 and σ r-8 dE) none of the above t. b. If the continuous random variable X is uniformly distributed witlh a mean of 45 and a variance of 75. The probability that X lies between 40 and 50 is: (C) 2/3 (D) 1/2 (E) none of the above t If a random variable X has a PDE fxCx) 0 elsewhere then EX) i (C) 4/5 (D) 5/4 (E) none of the above p t. d. If a random variable X has a PDF fx(x) 2(x 1) 1sxs2 elsewhere , then σ/s: VAR (E) none of the above e. Let X be a real-valued, continuous raudom variable and let X. Then, If y 0, then the (A) 1/20 ( 208 (D) 1/24 cumulative distribution function Fo) C DP

With explanation please.

Answer 1

a)

$\mu$y =E(2-2X)=2-2E(X)=2-$\mu$x =2-2*0 =2

$\sigma$y=SD(2-2X)=2*$\sigma$x=2*4 =8

option E is correct

b)

here as mean =(a+b)/2 =45

a+b=90........(1)

and variance =(b-a)²/12 =75

b-a =30 .........(2)

solving above equations ; a =30 ; b=60

therefore probability X lie between 40 and 50 =(50-40)/(60-30)=10/30=1/3

option A

c)

E(X)=$\int_{1}^{2}$x f(x) dx= $\int_{1}^{2}$2(x²-x) dx =2*(x³/3-x²/2)|²₁ =5/3

option B

d)

E(X²)=$\int_{1}^{2}$x² f(x) dx= $\int_{1}^{2}$2(x³-x²) dx =2*(x⁴/4-x³/3)|²₁ =17/6

hence $\sigma ^{2}$x =E(X²)-(E(X))² =1/18

option C:

e)

Cumulative funcrtion F(y) =P(Y<y)=P(X²<y) =P(-√y <X<√y) =$\int_{-\sqrt{y}}^{\sqrt{y}}$ f_x(x) dx

= F_x(x)|^√y_-√y   =F_x(√y)-F_x(-√y)

option II