If a variable takes values $0, 1, 2, ..., n$ with frequencies proportional to the binomial coefficients $^nC_0, ^nC_1, ..., ^nC_n$,find the mean of the distribution.

If a discrete random variable $X$ is defined as follows:
$P(X=x) = \begin{cases} \frac{k(x+1)}{5^x}, & x=0, 1, 2, \ldots \\ 0, & \text{otherwise} \end{cases}$
then $k=$

If the probability that an individual will suffer a bad reaction from an injection is $0.001$, then the probability that out of $2000$ individuals, exactly $3$ individuals suffer a bad reaction is

Suppose $X$ has the following probability mass function $P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3$. What is $E[X^2]$?

$A$ person throws an unbiased die. If the number shown is even,he gains an amount equal to the number shown. If the number is odd,he loses an amount equal to the number shown. Then his expectation is ₹.

An observer counts $240$ vehicles per hour at a specific location on a highway. Assuming that the arrival of vehicles at the location follows a Poisson distribution, the probability that more than two vehicles arrive over a $30 \text{ sec}$ time interval is