Statistics PYQs - Last 10 Years

A random variable $X$ has the following probability distribution :
$$ \begin{array}{|l|c|c|c|c|} \hline \mathrm{X}=x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(\mathrm{X}=x) & 0.1 & 0.2 & 0.3 & 0.4 \\ \hline \end{array} $$
The mean and standard d...

Let mean and standard deviation of probability distribution
$$ \begin{array}{|c|c|c|c|c|} \hline \mathrm{X}=x & -3 & 0 & 1 & \alpha \\ \hline \mathrm{P}(\mathrm{X}=x) & \frac{1}{4} & \mathrm{~K} & \frac{1}{4} & \frac{1}{3} \\ \hline \end{ar...

The probability distribution of a random variable X is given by$$ \begin{array}{|l|c|c|c|c|c|} \hline \mathrm{X}=x_i & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}\left(\mathrm{X}=x_i\right) & 0.4 & 0.3 & 0.1 & 0.1 & 0.1 \\ \hline \end{array} $$
...

The mean and the standard deviation of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by $q$, where $p \neq 0$ and $q \neq 0$. If the new mean and new standard deviation (s.d.) b...

If X is a random variable with distribution given below

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding...

The variance of 20 observations is 5 . If each observation is multiplied by 3 and then 8 is added to each number, then variance of resulting observations is

The variance of first 50 even natural numbers is

The mean and variance of seven observations are 8 and 16 respectively. If 5 of the observations are $2,4,10,12,14$, then the square root of product of remaining two observations is

The mean of $n$ observations is $\bar{x}$. If three observations $\mathrm{n}+1, \mathrm{n}-1,2 \mathrm{n}-1$ are added such that mean remains same, then value of $n$ is

A student scores the following marks in five tests : $54,45,41,43,57$. His score is not known for the sixth test. If the mean score is 48 in six tests, then the standard deviation of marks in six tests is

The mean of 100 observations is 50 and their standard deviation is 5 , then the sum of all squares of all the observations is

A random variable $X$ has the following probability distribution

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidde...

Consider three observations $\mathrm{a}, \mathrm{b}$ and c such that $b=a+c$. If the standard deviation of $\mathrm{a}+2, \mathrm{~b}+2, \mathrm{c}+2$ is d , then ............ holds.

If the sum of the deviations of 50 observations from 30 is 50 , then the mean of these observations is

The cumulative distribution function of a discrete random variable X is given by

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
...

The mean and variance of seven observations are 8 and 16 respectively. If five of the observations are $2,4,10,12,14$, then the product of remaining two observations is

The mean of the numbers $a, b, 8,5,10$ is 6 and the variance is $6.8$ . Then which of the following gives possible values of $a$ and $b$ ?

Variance of first n natural numbers is ________.

In an experiment with 15 observations for $x$, the following results were available $\sum x^2=2830, \sum x=170$. One observation 20 was found to be wrong and was replaced by the correct value 30 . Then the corrected variance is

 If for some $x \in \mathbb{R}^{+} \cup\{0\}$, the frequency distribution of the marks obtained by 20 students in a test is

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;fon...

Mean and variance of six observations are 6 and 12 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is

The mean and variance of 7 observations are 8 and 16 respectively. If first five observations are $2,4,10,12,14$, then absolute difference of remaining two observations is

The standard deviation of the following distribution

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10...

A random variable \(\mathrm{X}\) assumes values 1, 2, 3, ....., n with equal probabilities, if \(\operatorname{var}(X)=E(X)\), then \(\mathrm{n}\) is

If the mean and S.D. of the data \(3,5,7, a, b\) are 5 and 2 respectively, then \(a\) and \(b\) are the roots of the equation

The variance of 20 observations is 5. If each observation is multiplied by 2, then variance of resulting observations is

If both mean and variance of 50 observations \(x_1, x_2, \ldots, x_{50}\) are equal to 16 and 256 respectively, then mean of \(\left(x_1-5\right)^2,\left(x_2-5\right)^2, \ldots \ldots,\left(x_{50}-5\right)^2\) is

The raw data \(x_1, x_2, \ldots \ldots, x_{\mathrm{n}}\) is an A.P. with common difference \(\mathrm{d}\) and first term \(0, \bar{x}\) and \(\sigma^2\) are mean and variance of \(x_{\mathrm{i}}, \mathrm{i}=1,2, \ldots \ldots \mathrm{n}\), ...

The c.d.f. \(F(x)\) associated with p.d.f. \(f(x)\)
$$f(x)=\left\{\begin{array}{cl}12 x^2(1-x), & \text { if } 0< x <1 \\ 0 ; & \text { otherwise }\end{array}\right.$$ is

If the sum of the mean and the variance of a binomial distribution for 5 trials is 1.8 , then the value of \(p\) is

Variance of first \(2 n\) natural numbers is

If the variance of the numbers \(-1,0,1, \mathrm{k}\) is 5, where \(\mathrm{k} > 0\), then \(\mathrm{k}\) is equal to

For 20 observations of variable $x$, if \(\sum\left(x_i-2\right)=20\) and \(\sum\left(x_i-2\right)^2=100\), then the standard deviation of variable \(x\) is

The discrete random variable \(\mathrm{X}\) can take all possible integer values from 1 to \(\mathrm{k}\), each with a probability \(\frac{1}{\mathrm{k}}\), then its variance is

If variance of \(x_1, x_2 \ldots \ldots, x_n\) is \(\sigma_x^2\), then the variance of \(\lambda x_1, \lambda x_2, \ldots \ldots, \lambda x_{\mathrm{n}}(\lambda \neq 0)\) is

If both mean and variance of 50 observations \(x_1, x_2, \ldots \ldots, x_{50}\) are equal to 16 and 256 respectively, then mean of \(\left(x_1-5\right)^2,\left(x_2-5\right)^2, \ldots \ldots\left(x_{50}-5\right)^2\) is

The variance, for first six prime numbers greater than 5, is

Mean and variance of six observations are 8 and 16 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is

The variance and mean of 15 observations are respectively 6 and 10 . If each observation is increased by 8 then the new variance and new mean of resulting observations are respectively

The arithmetic mean of marks in Mathematics for four divisions A, B, C and D were \(80,75,70\) and 72 respectively. Their standard deviations were \(12,6,8\) and 10 respectively. Then, division ______ has more uniformity.

For the set of 50 observations, the sum of their squares is 3050 , their arithmetic mean is 6. Hence the standard deviation of these observations is

140. Given that total of 16 values is 528 and sum of the squares of deviation from 33 is 9158. The variance is

If the standard deviation of data is 12 and mean is 72, then coefficient of variation is

If the variance of the data 2, 4, 5, 6, 8, 17 is 23.33, then the variance of 4, 8, 10, 12, 16, 34 will be

Following data shows the information about marks obtained in Physics, Chemistry, Mathematics and Biology by 100 students in a class. Then subject shows the highest variability in marks

.tg {border-collapse:collapse;border-spacing:0;}
.tg ...

For two data sets each of size 5 , the variance are given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively. The variance of the combined data set is

A bakerman sells 5 types of cakes. Profit due to sale of each type of cake is respectively ₹ 2.5, ₹ 3 , ₹ 1.5 and ₹ 1. The demands for these cakes are \(20 \%, 5 \%, 10 \%, 50 \%\) and respectively, then the expected profit per cake is

For X ~ B(n, p), if p = 0.6, E(X) = 6, then Var(X) =

The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2 and 6, then the other two are

If 1 is added to first 10 natural numbers, then variance of the numbers so obtained is