$A$ coin is tossed twice in succession. Let $X$ represent the number of tails in two tosses,then the probability distribution of $X$ is given by

If $P(X=x)=k\left(\frac{3}{8}\right)^{X}, x=1,2,3, \ldots$ is the probability distribution function of a discrete random variable $X$,then $k=$

Let $X$ be the random variable taking values $1, 2, \ldots, n$ for a fixed positive integer $n$. If $P(X=k) = \frac{1}{n}$ for $1 \leq k \leq n$,then the variance of $X$ is

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.

If the function $f$ defined by $f(x) = \begin{cases} K(x-x^2) & \text{if } 0 < x < 1 \\ 0 & \text{otherwise} \end{cases}$ is the probability density function (p.d.f.) of a random variable $X$,then the value of $P(X < \frac{1}{2})$ is

Given the probability density function (p.d.f.) of the random variable $X$ as $f(x) = \frac{1}{2a}$ for $0 < x < 2a$ and $f(x) = 0$ otherwise, where $a > 0$, which of the following is correct?