If $X$ is a random variable with probability distribution $P(X=k) = \frac{(k+1)c}{2^k}$ for $k = 0, 1, 2, \ldots$,then $P(X \geq 3) = $

If the probability function of a random variable $X$ is given by $P(X=n) = \frac{k(n+1)}{3^n}$ for $n \in \mathbb{N} \cup \{0\}$ where $k$ is a constant, then $P(X < 2) = $

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 $k$.

Three rotten apples are accidentally mixed with fifteen good apples. Assuming the random variable $X$ to be the number of rotten apples in a draw of two apples,the variance of $X$ is

$A$ box contains $8$ red and $N$ green balls. Two balls are drawn at random from it. If $X$ is the random variable representing the number of green balls drawn and $E(X) = 1.2$, then $N = ...$

If the p.m.f. of a random variable $X$ is given by the following table,then the standard deviation of $X$ is (given $p+q=1$):

$x$	$0$	$1$	$2$
$P(X=x)$	$q^2$	$2pq$	$p^2$