If the mean of a Poisson distribution is $\frac{1}{3}$,then the ratio $P(X=1) : P(X=2)$ is

If a discrete random variable $X$ takes values $0, 1, 2, 3, \ldots$ with probability $P(X=x) = k(x+1) 5^{-x}$,where $k$ is a constant,then $P(X=0)$ is

If $x$ is a random variable with $PMF$ as follows: $P(X = x) = \begin{cases} \frac{5}{16}, & x = 0, 1 \\ \frac{kx}{48}, & x = 2 \\ \frac{1}{4}, & x = 3 \end{cases}$ then find $E(x)$.

If $X$ is a Poisson variate satisfying the condition $3 P(X=2)=P(X=4)$,then find $P(X=6)$.

$A$ random variable $X$ has the following probability distribution:
| $X$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $P(X)$ | $k^2$ | $2k$ | $k$ | $2k$ | $5k^2$ |
Then $P(X \geq 2)$ is equal to:

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