Question 1 : The cumulative distribution function of a random variable $x$ is defined as<br/>

Question 2 : A random variable X has its range $X = \{3, 2, 1\}$ with the probabilities,<br/>$\dfrac{1}{2},\dfrac{1}{3}$ and $\dfrac{1}{6}$ respectively. The mean value of X is<br/>

Question 3 : <p><span>10 different books and 2 different pens are given to 3 boys so that each gets equal number of things. The probability that the same boy does not receive both the pens is</span></p>

Question 4 : An urn contains $5$ red and $2$ black balls. Two balls are randomly drawn. Let $X$ represent the number of black balls. What are the possible values of $X$ ? Is $X$ a random variable ?

Question 5 : In the following, find the value of k.<br/>$P(x)=\left\{\begin{matrix} kx &amp; for &amp; x=1, 2, 3\\ 0 &amp; for &amp; otherwise\end{matrix}\right.$.<br/>

Question 6 : If the range of the random variable X is from $a$ to $b$ then $F(X\leq b)$ is<br>

Question 7 : If a random variable X takes value $ 0 $ and $1$ with respective probabilities $\dfrac{2}{3}$ and $\dfrac{1}{3}$ , then the expected value of X is<br/>

Question 8 : The probability distribution of a discrete random variable $X$ is:<table class="wysiwyg-table"><tbody><tr><td>$X = x$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>$P(X = x)$</td><td>$k$</td><td>$2k$</td><td>$3k$</td><td>$4k$</td><td>$5k$</td></tr></tbody></table>Find $P (X\leq 4)$

Question 9 : In the p.d.f. of a random variable of three missing entries are in the ratio $1:2:3$.<table class="wysiwyg-table"><tbody><tr><td>$X=x$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>$P(X=x)$</td><td>$\dfrac{1}{10}$</td><td>-</td><td>-</td><td>-</td><td>$\dfrac{3}{20}$</td></tr></tbody></table>Then the missing entries are?<br>

Question 10 : Expected number of heads when we toss $n$ unbiased coins is<br>

Question 11 : A box contains 3 white and 2 black balls. Two balls are drawn at random one after the other. If the balls are not replaced. what is the probability that both the balls are black ?<span><br/></span>

Question 12 : Four different objects 1,2,3,4 are distributed at random in four places marked 1,2,3,4. What is the probability that none of the objects occupy the place corresponding to its number ?

Question 13 : In $5$ throws of a die, getting $1$ or $2$ is a success. The mean number of successes is

Question 14 : The probability distribution of a discrete random <span>variable X is given in the following table :<br></span><table class="wysiwyg-table"><tbody><tr><td>$X = x$</td><td>$0$</td><td>$1$</td><td>$2$</td></tr><tr><td>$P(x)$</td><td>$4C^{3}$</td><td>$4C - 13C^{2}$</td><td>$7C - 1$</td></tr></tbody></table><span>; $C &gt; 0$ then $C =$ ________.</span>

Question 15 : If the range of the random variable X is from a to $\mathrm{b},\ \mathrm{a}&lt;\mathrm{b},\ \mathrm{F}(\mathrm{X}&lt;\mathrm{a})=$<br>

Question 16 : In a class of 80 students, 48 are boys and the rest of the students are girls. If 10 students shift to the other class room and a student is selected at random from the remaining class, what is the probability that a girl is selected ?

Question 17 : A player tosses two fair coins. He wins $Rs.\ 5/-$ if two heads occur, $Rs.$ $2/-$ if one head occurs and $Rs.$ $1/-$ if no head occurs. Then his expected gain is<br>

Question 18 : The expected value of the number of points, obtained in a single throw of die, is

Question 20 : A random variable $X$ has the following probability mass function:<br/><table class="wysiwyg-table"><tbody><tr><td>$X$</td><td>$-2$</td><td>$3$</td><td>$1$</td></tr><tr><td>$P(X = x)$</td><td>$\dfrac{\lambda}{6}$</td><td>$\dfrac{\lambda}{4}$</td><td>$\dfrac{\lambda}{12}$</td></tr></tbody></table>Then the value of $\lambda$ is:

Question 21 : The probability function of a binomial distribution is $P(x) = \binom{6}{x} p^{x} q^{6 - x}, x = 0, 1, 2, ..., 6$. If $2P(2) = 3P(3)$, then $p =$ __________.

Question 22 : The mean or average number of points when we throw a symmetrical die is<br/>

Question 23 : For the probability distribution given by $\left.\begin{matrix} X=x_i &amp; 0 \\ P. &amp; \dfrac{25}{36}\end{matrix}\right|$ $\begin{matrix} 1 \\ 5 \\ 18\end{matrix}$ $\begin{vmatrix} 2 \\ 1 \\ 36\end{vmatrix}$ the standard deviation $(\sigma)$ is?

Question 24 : A random variable X follows the following distribution <br/>$X=x_{i}: \quad \ \ 1 , \ 2 , \ 3, \ 4 $<div>$p(X=x_{i}):\dfrac{2}{6}, \dfrac{3}{6} , \dfrac{0}{6}, \dfrac{1}{6}$<br/>, then the mean and variance are <br/></div>

Question 25 : The outcome of each of $30$ items was observed; $10$ items gave an outcome $\dfrac{1}{2}$- d each, $10$ items gave outcome $\dfrac{1}{2}$ each and the remaining $10$ items gave outcome $\dfrac{1}{2}+$ d each. If the variance of this outcome data is $\dfrac{4}{3}$ then $|d|$ equals:-

Question 26 : <span>If $X$ is a random variable with the following probability distribution given below:</span><div><table class="table table-bordered"><tbody><tr><td> $X=x$</td><td> $0$</td><td>$1$ </td><td>$2$ </td><td>$3$ </td></tr><tr><td>$P(X=x)$ </td><td> $k$</td><td>$3k$ </td><td>$3k$ </td><td>$k$</td></tr></tbody></table><span>Then the value of $k$ and its variance are:</span></div>

Question 27 : The probability distribution of a random variable $X$ is given below, then $K =$<br/>$X=x_{1}: \quad 1 \quad , 2, \quad 3, \quad 4$<br/>$p(X=x_{1}):2k , 4k \ \ ,3k , \ \ \ k$<br/>

Question 28 : A coin is rolled n times. If the probability of getting head at least once is greater than $90\%$ then the minimum value of n is?

Question 29 : A random variable $X$ has the following probability distribution:<br/><table class="wysiwyg-table"><tbody><tr><td>$X$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>$P(X)$</td><td>$k^{2}$</td><td>$2k$</td><td>$k$</td><td>$2k$</td><td>$5k^{2}$</td></tr></tbody></table>Then $P(X&gt;2)$ is equal to

Question 30 : Two cards are drawn simultaneously from a well shuffled pack of $52$ cards. The expected number of aces is<br/>

Question 31 : What is the mean of $f(x)=3x+2$ where x is a random variable with probability distribution.<table class="wysiwyg-table"><tbody><tr><td>$X=x$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td></tr><tr><td>$P(X=x)$</td><td>$1_{/6}$</td><td>$1_{/3}$</td><td>$1_{/3}$</td><td>$1_{/6}$</td></tr></tbody></table>

Question 32 : The mathematical expectation of sum of points when we throw n symmetrical dice is<br/>

Question 33 : <span>A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required. </span>The probability that $X\geq 3$ equals<br/>