1. A random variable X may have no moments but although its m.g.f exist- justify the statement.
a. True
b. False
c. not enough information.
Ans: a
2. If Z=\frac{X-\mu}{\sigma }, then find E(Z)
a. 0
b. \mu
c. \sigma
Ans: a
3. What is the relation between the fourth order central moment and cumulants.
a. \mu _{4}=\kappa_{4}+3\kappa_{2}^{2}
b. \mu _{4}=\kappa_{2}+3\kappa_{2}^{2}
c. \mu _{4}=\kappa_{2}+3\kappa_{4}^{2}
Ans: a
4. According to characteristics function \phi (0)=?
a. 0
b.1
c.\infty
Ans:b
5."Characteristics function follows the property of conjugate function" - is it true or flase?
a. False
b. True
Ans: b
6. A symmetric die is thrown 600 times. find the probability(lower bound) of getting 80 to 120 sixes.
a. \frac{19}{24}
b. \frac{15}{24}
c. \frac{10}{24}
Ans: a
7. If X has p.g.f P(S). Then X+1 has __________________.
a. sP(s)
b. s^2P(s)
c. s^3P(s)
Ans: a
8. If X_{n}\overset{p}{\rightarrow}\alpha and Y_{n}\overset{p}{\rightarrow}\beta ,
n \rightarrow \infty. Then X_{n}\pm Y_{n}\overset{p}{\rightarrow} =? as n \rightarrow \infty.
a. \frac{\alpha}{\beta}
b. \alpha \pm \beta
c. \alpha \beta
9.\lim_{t\rightarrow \infty}\phi (t)=?
a. 0
b. 1
c. \infty
Ans: a
10. m.g.f holds the Uniqueness theorem. Is it true?
a. No
b. Yes
c. Info not enough
Ans: b
Do it your self:
1.Find the m.g.f for for geometric distribution. Also find the mean and the variance of it.
2.If the moments of the variate X are defined by E(X^r)=0.6; r=1,2,3,..... show that P(X=0)=0.4, P(X=1)=0.6, P(X\geqslant 2)=0.
3.Two unbiased dices are thrown. If X is the sum of the numbers showing up, proven that P(|X-7|\geqslant 3)\leqslant \frac{35}{54}.
4.If P{\left | \bar{X}_{n}-\mu \right |<0.5}\geqslant 0.95, then find the the sample size by chebychev's inequality?
5.If p(x)=\frac{1}{2^x}; x=1,2,3.......... Find the mgf, mean, var.