$A$ random variable $X$ has the following probability distribution:
$X$: $1, 2, 3, 4$
$P(X)$: $0.2, 0.4, 0.3, 0.1$
The mean and variance of $X$ are respectively:

$A$ random variable $X$ has the range $\{0, 1, 2, \ldots\}$. If $P(X=r) = k(1+r) 3^{-r}$ for $r=0, 1, 2, \ldots$, where $k > 0$ is a real number, then $P(X=0) + P(X=1) + P(X=2) =$

$A$ random variable $X$ takes values $0, 1, 2, 3, \ldots$ with probability $P(X=x) = K(x+1)\left(\frac{1}{5}\right)^x$,where $K$ is a constant. Then $P(X=0)$ is:

Let $X$ be a random variable with probability distribution function,$P(X=x)=K\left(\frac{2}{5}\right)^x, x=1, 2, 3, \ldots$. Then,the value of $K$ is

In a game,a man wins $₹ 40$ if he gets $5$ or $6$ on a throw of a fair die and loses $₹ 20$ for getting any other number on the die. If he decides to throw the die either until he gets a $5$ or $6$ or to a maximum of $3$ throws,then his expected gain/loss (in rupees) is:

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.