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) = $

$A$ player tosses $2$ fair coins. He wins ₹ $5$ if $2$ heads appear,₹ $2$ if one head appears,and ₹ $1$ if no head appears. Then the variance of his winning amount in ₹ is

The probability distribution of a random variable $X$ is given by the following table:

$X = x_i$	$3$	$5$	$7$	$9$
$P(X = x_i)$	$k$	$2k$	$3k$	$4k$

Then the standard deviation of $X$ is

$A$ person draws a card from a pack of $52$ cards and replaces it. After shuffling,he draws a card again. If he repeats this process,what is the probability that he draws a red card for the first time on the third draw?

Given the probability density function: $f(x) = \begin{cases} 3(1 - 2x^2), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}$ The probability $P\left(\frac{1}{4} < X < \frac{1}{3}\right)$ is given by: $P\left(\frac{1}{4} < X < \frac{1}{3}\right) = \int_{1/4}^{1/3} 3(1 - 2x^2) \, dx$

If $X$ is a Poisson variate such that $P(X=1) = 2P(X=2)$,then $P(X=3)$ is equal to: