The p.m.f of a random variable $X$ is given by $P(X) = \frac{2x}{n(n+1)}$ for $x = 1, 2, 3, \ldots, n$,and $0$ otherwise. Then $E(X) = $

If $X$ is a Poisson variate with $P(X=0)=0.8$,then the variance of $X$ is

Suppose that a random variable $X$ follows a Poisson distribution. If $P(X=1) = P(X=2)$,then $P(X=5)$ is equal to:

$A$ random variable $X$ has the following probability distribution. For events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 4\}$,what is the probability $P(E \cup F)$?

$X$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$P(X)$	$0.15$	$0.23$	$0.12$	$0.10$	$0.20$	$0.08$	$0.07$	$0.05$

The probability distribution of a random variable $X$ is given below. Then,the standard deviation of $X$ is

$X=x_i$	$2$	$3$	$5$	$7$	$12$
$P(X=x_i)$	$3k$	$k$	$k$	$2k$	$k$

The cumulative distribution function (c.d.f.) $F(x)$ associated with the probability density function (p.d.f.) $f(x) = 3(1 - x^2)$ for $0 < x < 1$ and $f(x) = 0$ otherwise,is given by $F(x) = k(x - \frac{2x^3}{k})$. Find the value of $k$.