Vinod Singh and Shweta Pawar Solutions for Chapter: Probability Distribution, Exercise 2: Critical Thinking

The probability mass function of a random variate $X$ is $P\left(x\right)=\left\{\begin{array}{cl}k{x}^2,& x=1, 2, 3, 4\\ 0,& \mathrm{otherwise}\end{array}\right.$. Then, $E\left(X\right)=$

If the probability mass function of a discrete random variable $X$ is $P\left(x\right)=\left\{\begin{array}{ll}\frac{C}{x^3},& x=1, 2, 3\\ 0,& \text{otherwise}\end{array}\right.$. Then $E\left(X\right)=$

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

Then, $4E\left(X^2\right)-\mathrm{Var}\left(X\right)$ is equal to

A random variable $X$ has the following probability mass function

If $p+q=1$, then the variance of $X$ is

The probability mass function of a random variable $X$ is given by

If three fair coins are tossed, where $X=$number of heads obtained, then $E\left(X\right)$ is

If a coin is tossed twice and $X$ is the number of tails, then $\mathrm{Var}\left(X\right)=$

Two dice are thrown simultaneously. If $X$ denotes the number of sixes, then the expected value of $X$ is