The p.m.f. of a random variable $X$ is $P(x) = \begin{cases} \frac{2x}{n(n+1)}, & x = 1, 2, 3, \ldots, n \\ 0, & \text{otherwise} \end{cases}$,then $E(X)$ is

An executive in a company makes on an average $5$ telephone calls per hour at a cost of $Rs. 2$ per call. The probability that in any hour the cost of the calls exceeds a sum of $Rs. 4$ is

$A$ player tosses two coins. He wins $Rs. 1$ if $1$ head appears,$Rs. 2$ if $2$ heads appear. But he loses $Rs. 3$ if no head appears. The mean of the prize money is

If a random variable $X$ has the following probability distribution of $X$:

$X=x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X=x)$	$0$	$k$	$2k$	$2k$	$3k$	$k^2$	$2k^2$	$7k^2+k$

Then $P(X \geqslant 6) = $

$A$ person plays a game of tossing a coin thrice. For each head,he is given $Rs. 2$ by the organiser of the game and for each tail,he has to give $Rs. 1.50$ to the organiser. Let $X$ denote the amount gained or lost by the person. Show that $X$ is a random variable and exhibit it as a function on the sample space of the experiment.

If a drunkard person tries to take a step,then it will be a forward or backward step with probabilities $\frac{1}{4}$ and $\frac{1}{2}$ respectively,or he will remain in his 'as it is' position. If he tries to take a step $5$ times,then find the probability that he will be one step away from the initial position.