Probability & Statistics PYQ — Page 6

If the mean and variance of the frequency distribution <table class="pyq-table"><tbody><tr><td>${x}_{i}$</td><td>$2$</td><td>$4$</td><td>$6$</td><td>$8$</td><td>$10$</td><td>$12$</td><td>$14$</td><td>$16$</td></tr><tr><td>${f}_{i}$</td><td>$4$</td><td>$4$</td><td>$\alpha$</td><td>$15$</td><td>$8$</td><td>$\beta$</td><td>$4$</td><td>$5$</td></tr></tbody></table>are $9$ and $15.08$ respectively, then the value of ${\alpha }^{2}+{\beta }^{2}-\alpha \beta$ is _____.

The mean and variance of $5$ observations are $5$ and $8$ respectively. If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is

Let the mean of the data <table class="pyq-table"><tbody><tr><td>$x$</td><td>$1$</td><td>$3$</td><td>$5$</td><td>$7$</td><td>$9$</td></tr><tr><td>Frequency $(f)$</td><td>$4$</td><td>$24$</td><td>$28$</td><td>$\alpha$</td><td>$8$</td></tr></tbody></table>be $5.$ If $m$ and ${\sigma }^{2}$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3\alpha }{m+{\sigma }^{2}}$ is equal to $_______.$

If the probability that the random variable $X$ takes values $x$ is given by $P(X=x)=k(x+1){3}^{-x}$, $x=0,1,2,3,\ldots \ldots$, where $k$ is a constant, then $P(X\geq 2)$ is equal to

Let $\Omega$ be the sample space and $A\subseteq \Omega$ be an event. Given below are two statements: (S1): If $P(A)=0$, then $A=\phi$ (S2): If $P(A)=$, then $A=\Omega$ Then

Let $A$ be the event that the absolute difference between two randomly chosen real numbers in the sample space $[0,60]$ is less than or equal to $a$. If $P(A)=\frac{11}{36}$, then a is equal to _____ .

If an unbiased die, marked with $-2,-1,0,1,2,3$ on its faces is thrown five times, then the probability that the product of the outcomes is positive, is :

Let the six numbers ${a}_{1},{a}_{2},...,{a}_{6}$ be in $A.P.$ and ${a}_{1}+{a}_{3}=10$.If the mean of these six numbers is $\frac{19}{2}$ and their variance is ${\sigma }^{2}$, then $8{\sigma }^{2}$ is equal to

Let M be the maximum value of the product of two positive integers when their sum is $66$ . Let the sample space $S={x\in Z:x(66-x)\geq \frac{5}{9}M}$ and the event $A={x\in S:x$ is a multiple of $3$}. Then $P(A)$ is equal to

Let the probability of getting head for a biased coin be $\frac{1}{4}$. It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64{x}^{2}+5Nx+1=0$ has no real root is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q-p$ is equal to..........

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution <table class="pyq-table"><tbody><tr><td>${X}_{i}$</td><td>$0$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>${f}_{i}$</td><td>$k+2$</td><td>$2k$</td><td>${k}^{2}-1$</td><td>${k}^{2}-1$</td><td>${k}^{2}+1$</td><td>$k-3$</td></tr></tbody></table>where $\Sigma {f}_{i}=62$. If $[x]$ denotes the greatest integer $\leq x$, then$[{\mu }^{2}+{\sigma }^{2}]$ is equal to

Let $X={11,12,13,\ldots .,40,41}$ and $Y={61,62,63,...,90,91}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and ${\sigma }^{2}$ is the variance of all the observations in $X\cup Y$, then $|\bar{x}+\bar{y}-{\sigma }^{2}|$ is equal to ________

$25%$ of the population are smokers. A smoker has $27$ times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$.Then the value of $k$ is _____ .

In a bolt factory, machines $A,B$ and $C$ manufacture respectively $20%,30%$ and $50%$ of the total bolts. Of their output $3,4$ and $2$ percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective then the probability that it is manufactured by the machine $C$ is

Let $S={M=[{a}_{ij}],{a}_{ij}\in {0,1,2},{1\leq i,j\leq 2}}$ be a sample space and $A{M\in S:M\text{is invertible}}$ be an even. Then $P(A)$ is equal to

The mean and variance of $7$ observations are $8$ and $16$ respectively. If one observation $14$ is omitted, $a$ and $b$ are respectively mean and variance of remaining $6$ observation, then $a+3b-5$ is equal to ________

Let the mean and standard deviation of marks of class A of $100$ students be respectively $40$ and $\alpha (>0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively $55$ and $30-\alpha$. If the mean and variance of the marks of the combined class of $100+n$ students are respectively $50$ and $350$ , then the sum of variances of classes $A$ and $B$ is

A bag contains $6$ white and $4$ black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is

Let the mean and variance of $12$ observations be $\frac{9}{2}$ and $4$ respectively. Later on, it was observed that two observations were considered as $9$ and $10$ instead of $7$ and $14$ respectively. If the correct variance is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ are coprime, then $m+n$ is equal to

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$. If $p:q=m$ $:n$, where $m$ and $n$ are co-prime, then $m+n$ is equal to

The mean and variance of a set of $15$ numbers are $12$ and $14$ respectively. The mean and variance of another set of $15$ numbers are $14$ and ${\sigma }^{2}$ respectively. If the variance of all the $30$ numbers in the two sets is $13$, then ${\sigma }^{2}$ is equal to

If the mean of the frequency distribution <table class="pyq-table"><tbody><tr><td>Class :</td><td>$0-10$</td><td>$10-20$</td><td>$20-30$</td><td>$30-40$</td><td>$40-50$</td></tr><tr><td>Frequency :</td><td>$2$</td><td>$3$</td><td>$x$</td><td>$5$</td><td>$4$</td></tr></tbody></table>is $28$, then its variance is ________ .

Let sets $A$ and $B$ have $5$ elements each. Let the mean of the elements in sets $A$ and $B$ be $5$ and $8$ respectively and the variance of the elements in sets $A$ and $B$ be $12$ and $20$ respectively. A new set $C$ of $10$ elements is formed by subtracting $3$ from each element of $A$ and adding $2$ to each element of $B$. Then the sum of the mean and variance of the elements of $C$ is

The mean and variance of the marks obtained by the students in a test are $10$ and $4$ respectively. Later, the marks of one of the students is increased from $8$ to $12$ . If the new mean of the marks is $10.2$. then their new variance is equal to: