$A$ boy tosses a fair coin $3$ times. If he gets $₹ 2x$ for $x$ heads,then his expected gain equals to $₹........$

If a discrete random variable $X$ is defined as follows:
$P(X=x) = \begin{cases} \frac{k(x+1)}{5^x}, & x=0, 1, 2, \ldots \\ 0, & \text{otherwise} \end{cases}$
then $k=$

An unbiased die is tossed until a number greater than $4$ appears. The probability that an even number of tosses is needed is

In a city,$10$ accidents take place in a span of $50$ days. Assuming that the number of accidents follows the Poisson distribution,the probability that three or more accidents occur in a day is:

The discrete random variable $X$ can take all possible integer values from $1$ to $k$,each with a probability $\frac{1}{k}$. Then its variance is

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 $k$.