Find the mean number of heads in three tosses of a fair coin. (in $.5$)

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

The probability distribution of a random variable $X$ is given by

$X=x_i$	$0$	$1$	$2$	$3$	$4$
$P(X=x_i)$	$0.4$	$0.3$	$0.1$	$0.1$	$0.1$

Then the variance of $X$ is

$A$ die is rolled. If $X$ denotes the number of positive divisors of the outcome,then the range of the random variable $X$ is

If $X$ is a Poisson variate such that $P(X=1)=P(X=2)$,then $P(X=4)$ is equal to

If the probability distribution of a discrete random variable $X$ is given by $P(X=k) = \frac{2^{-k}(3k+1)}{2^c}$ for $k = 0, 1, 2, \ldots, \infty$,then find the value of $P(X \leq c)$. Note: The expression provided in the prompt is likely $P(X=k) = \frac{(3k+1)}{2^{k+c}}$. Given $\sum_{k=0}^{\infty} P(X=k) = 1$,we have $\frac{1}{2^c} \sum_{k=0}^{\infty} (3k+1) \left(\frac{1}{2}\right)^k = 1$.