The probability distribution of a random variable $X$ is given by

$X=x$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$0.4$	$0.3$	$0.1$	$0.1$	$0.1$

The variance of $X$ is

$A$ 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.

For the following probability distribution,find the expected value of $X$:

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$P(x)$	$k$	$0.3$	$0.15$	$0.15$	$0.1$	$2k$

Three rotten apples are mixed accidentally with seven good apples and four apples are drawn one by one without replacement. Let the random variable $X$ denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent the mean and variance of $X$,respectively,then $10(\mu^2 + \sigma^2)$ is equal to

Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0) = \frac{1}{2}$ and $P(X=j) = \frac{1}{3^j}$ for $j = 1, 2, 3, \ldots, \infty$. Then the mean of the distribution and $P(X \text{ is positive and even})$ respectively are:

Let $X$ represent the difference between the number of heads and the number of tails obtained when a coin is tossed $6$ times. What are the possible values of $X$?