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.