$A$ discrete random variable $X$ takes values $10, 20, 30,$ and $40$,with probabilities $0.3, 0.3, 0.2,$ and $0.2$ respectively. Then the expected value of $X$ is

The probability function of a random variable $X$ is given by $P(X=k)=c k^2$,where $c$ is a constant and $k \in\{0,1,2,3,4\}$. If $\sigma^2$ is the variance of $X$ and $\mu$ is the mean of $X$,then $\sigma^2+\mu^2=$

The probability distribution of a random variable $X$ is given below:

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X=x)$	$a$	$a$	$a$	$b$	$b$	$0.3$

If the mean of $X$ is $4.2$, then $a$ and $b$ are respectively equal to:

$A$ random variable $X$ has the probability distribution given below. Its variance is:

$X$	$1$	$2$	$3$	$4$	$5$
$P(X=x)$	$K$	$2K$	$3K$	$2K$	$K$

The cumulative distribution function $F(X)$ of a discrete random variable $X$ is given by the following table:

$X$	$1$	$2$	$3$	$4$	$5$	$6$
$F(X=x)$	$0.2$	$0.37$	$0.48$	$0.62$	$0.85$	$1$

Then $P[X=4] + P[X=5] = $

Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?

Outcome	Probability
$\omega_{1}$	$1/8$
$\omega_{2}$	$2/3$
$\omega_{3}$	$1/3$
$\omega_{4}$	$1/3$
$\omega_{5}$	$-1/4$
$\omega_{6}$	$-1/3$