Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 3: EXERCISE

The values of a random variable must be positive.

If the spectrum of a random variable $X$ is the set $\left\{0, 1, 2\right\}$, then $\mathrm{P}\left(\mathrm{X}\ge 0\right)=\mathrm{P}\left(\mathrm{X}\le 2\right)$.

If a random variable $X$ takes only four values $2, 3, 4, 5$ and if at these points the probabilities of $X$ are $0.1, 0.2, 0.6, 0.1$ respectively, then $\mathrm{P}\left(2\le X\le 5\right)=0.9$.

A random variable $\mathrm{X}$ is a real value function defined on the sample space of a random experiment.

For a random variable $\mathrm{X}$, the following system of numbers

$\begin{array}{|ccccc|}\hline \mathrm{X}& 0& 1& 2& 3\\ \mathrm{P}\left(\mathrm{X}\right)& 3k-1& 2k& 3k& k\\ \hline\end{array}$

is a probability distribution.

Two dice are thrown. If $\mathrm{X}$ denotes the sum of the numbers appeared on the dice thrown, the probability that $\mathrm{X}$ lies in the interval $\left(10, 12\right)$ is same as that of the event $\mathrm{E}=$

The probability distribution of a discrete random variable  is given by

$P(X=x)=\left\{\begin{array}{ll}2k& \text{ if }x=-1\\ k& \text{ if }x=0\\ 3k& \text{ if }x=1\\ 0,& \text{ otherwise }\end{array}\right.$

Then, the value of $k$ is

The probability distribution of a random variable $\mathrm{X}$ is given by

$P(X=x)=\left\{\begin{array}{ll}3k^2& \text{ if }x=0\\ 5k^2+k& \text{ if }x=1\\ 7k^2+k& \text{ if }x=2\\ 0,& \text{ otherwise }\end{array}\right.$

Then