If the probability function of a random variable $X$ is given by $P(X=j) = \frac{1}{2^j}$ for $j = 1, 2, 3, \ldots, \infty$,then the variance of $X$ is:

In a book of $500$ pages, it is found that there are $250$ typing errors. Assume that Poisson law holds for the number of errors per page. Then, the probability that a random sample of $2$ pages will contain no error, is:

Given below is the distribution of a random variable $X$:

$X=x$	$1$	$2$	$3$	$4$
$P(X=x)$	$\lambda$	$2\lambda$	$3\lambda$	$4\lambda$

If $\alpha=P(X < 3)$ and $\beta=P(X>2)$,then $\alpha: \beta=$

If a discrete random variable $X$ takes the values $1, 2, 3, 4$ such that $2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4)$, then $P(X = 4) = ...$ (in $61$)

Two dice are thrown simultaneously. If $X$ denotes the number of sixes,find the expectation of $X$.

The p.d.f. of a random variable $x$ is given by $f(x) = \frac{1}{4a}$ for $0 < x < 4a$ $(a > 0)$ and $f(x) = 0$ otherwise. If $P(x < \frac{3a}{2}) = k P(x > \frac{5a}{2})$,then $k = . . . . . .$