If the following function is a probability density function of a random variable $X$, $f(x) = kx^2(1 - x)$ for $0 < x < 1$ and $f(x) = 0$ otherwise, then the value of $k$ is:

$A$ die is loaded in such a way that each odd number is twice as likely to occur as each even number. If $E$ is the event that a number greater than or equal to $4$ occurs on a single toss of the die,then $P(E)$ is equal to:

Let $X$ denote the number of hours you study on a Sunday. It is known that $P(X=x) = \begin{cases} 0.1 & \text{if } x=0 \\ kx & \text{if } x=1, 2 \\ k(5-x) & \text{if } x=3, 4 \\ 0 & \text{otherwise} \end{cases}$ where $k$ is a constant. Then the probability that you study at least two hours on a Sunday is

If three fair coins are tossed,then the variance of the number of heads obtained is

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

Let $X$ be a random variable,and let $P(X=x)$ denote the probability that $X$ takes the value $x$. Suppose that the points $(x, P(X=x))$ for $x=0,1,2,3,4$ lie on a fixed straight line in the $xy$-plane,and $P(X=x)=0$ for all $x \in \mathbb{R} \setminus \{0,1,2,3,4\}$. If the mean of $X$ is $\frac{5}{2}$,and the variance of $X$ is $\alpha$,then the value of $24\alpha$ is: