Let a random variable $X$ take values $\{0, 1, 2, 3\}$ with $P(X=0) = P(X=1) = p$,$P(X=2) = P(X=3) = q$,and $E(X^2) = 2E(X)$. Then the value of $8p - 1$ is:

$A$ biased coin with probability $p, 0 < p < 1,$ of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $\frac{2}{5},$ then $p = $

If the probability that the random variable $X$ takes values $x$ is given by $P(X = x) = k(x + 1)3^{-x}$ for $x = 0, 1, 2, 3, \ldots$,where $k$ is a constant,then $P(X \geq 2)$ is equal to

In a communication network, $98 \%$ of messages are transmitted with no error. If a random variable $X$ denotes the number of incorrectly transmitted messages, then the probability that at most one message is transmitted incorrectly out of $500$ messages sent, is

If $P(X=2)=0.3, P(X=3)=0.4, P(X=4)=0.3$,then the variance of random variable $X$ is (in $.6$)

The p.d.f. of a random variable $X$ is given by $f(x) = \frac{k}{\sqrt{x}}$ for $0 \leq x \leq 4$ and $f(x) = 0$ otherwise. Then $P(1 < X < 4) = $