Suppose P(X-=0) = 1/4, P(X=1) = 3/4, Then X is a Bernoulli rv. a) true,b) false

Question

Suppose P(X-=0) = 1/4, P(X=1) = 3/4, Then X is a Bernoulli rv. a) true,b) false 2. In 1., E{X} = a) 1/4,b) 1/2,c) 3/4,d) 1,e) none of these 3. In 1., Var{X} = a) 3/4,b) 3/8,c) 3/16,d)3/64,e) none of these 4. In 1., E{X^4} = a) 0,b) 1/4,c) 3/4,d) 1,e) none of these 5. In 1., E{sin(2piX)} = a) 0,b) 1,c) 1/pi,d) pi,e) none of these 6. If X to Binomial(n=100,p=1/2), E(x) = a) 25,b) 50,c) 75,e) 100,e) none of these 7. If X to Binomial(n=100,p=1/2), Var(x) = a) 25,b) 50,c) 75,e) 100,e) none of these 8. If X is continuous uniform[0,100], then E(x) = a)25,b) 50,c) 75,e) 75,e) none of these 9. If X is continuous uniform[0,100], then Var(X) = a)10/12,b) 100/12,c) 1000/12,d) 10000/12,e) none of these 10. St.Dev(X) = squareroot (Var(X)). a) true,b) false 11. Let (x) = exp {-x^2/2}/squareroot2pi. Then integral_-infinity^infinity x (x) dx = a) 0,b) 1,c) 2,d) 3,e) none of these 12. In 11 integral_-infinity^infinity (x) dx = a) 0,b) 1,c) 2,d) 3,e) none of these 13. If Z Is N(0,1), then W = Z^2 is chisquare(1). a) true,b) false 14. If Z is N(0,1), then E{Z^3} = a) 0,b) 1,c) 2,d) 3,e) none of these 15. If X has pdf f(x) = (1/beta) exp(-x/beta) for x > 0. and 0 If x 0, and 0 if x

Explanation / Answer

1. Yes it is binomial Random variable, with q= 1/4 and p=3/4

2. In 1. E(X) =0* 1/4 + 1*3/4 = 3/4

3. In 1. Var(X) =E(X2) - E(X)2 = 02 * 1/4 + 12 * 3/4 - (3/4)2 = 3/16

4. E(X4) = 04 * 1/4 + 14 * 3/4 =3/4

5.E(sin 2*PI*X) = sin( 2* pi *0) *1/4 + sin(2*pi*1) *3/4 = 0

6.X is binomial n=100 p=1/2

E(X) = n*p = 50

7.X is binomial n=100 p=1/2

vAR(X) =n*p*q = 25

8. X is uniform [0,100]

E(X) = 1/2 * (0+100) = 50

9 X is uniform [0,100]

Var(X) = (100-0)2 / 12 = 10000/12

10 . S.d.2   = Var Hence True

11. The given expression is a pdf of standard normal distribution

And the function in the integral gives E(X) =mean =0

12. Phi(X) is pdf of standard normal distribution.

Integrating it over its domain gives, 1

13.If Z is normal then W= Z2 follows Chi-squared Distribution

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