NCERT Solutions for Chapter: Probability, Exercise 1: EXERCISE

For the following probability distribution determine standard deviation of the random variable $\mathrm{X}$.

A biased die is such that $\mathrm{P}\left(4\right)=\frac{1}{10}$ and other scores being equally likely. The die is tossed twice. If $\mathrm{X}$ is the 'number of four seen', find the variance of the random variable $\mathrm{X}$.

A die is thrown three times. Let $\mathrm{X}$ be the number of twos seen. Find the expectation of $\mathrm{X}$.

Two biased dice are thrown together. For the first die $\mathrm{P}\left(6\right)=\frac{1}{2},$ the other scores being equally likely while for the second die, $\mathrm{P}\left(1\right)=\frac{2}{5}$ and the other scores are equally likely. Find the probability distribution of " the number of ones seen ".

A factory produces bulbs. The probability that any one bulb is defective is $\frac{1}{50}$ and they are packed in boxes of $10 .$ From a single box, the probability that exactly two bulbs are defective is $45\times {\left(\frac{1}{50}\right)}^{10}\times k^8$. Find $k$.

Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.

The random variable $\mathrm{X}$ can take only the values of $0, 1, 2$. Given that $\mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)=\mathrm{p}$ and that $\mathrm{E}\left({\mathrm{X}}^2\right)=\mathrm{E}\left[\mathrm{X}\right]$, find the value of $\mathrm{p}$.

Let $X$ be a random variable taking values $x_1, x_2,\dots , x_n$ with probabilities $p_1, p_2,\dots , p_n$, respectively, then $\mathrm{Var}\left(X\right)=$$\_\_\_\_\_$.