JEE Main 2012 — Probability & Statistics

Let $\mathrm{x}_1, \mathrm{x}_2, \ldots \ldots, \mathrm{x}_{\mathrm{n}}$ be $\mathrm{n}$ observations, and let $\overline{\mathrm{x}}$ be their arithematic mean and $\sigma^2$ be their variance. Statement $1$: Variance of $2 x_1, 2 x_2, \ldots \ldots, 2 x_n$ is $4 \sigma^2$. Statement $2$: Arithmetic mean of $2 \mathrm{x}_1, 2 \mathrm{x}_2, \ldots . ., 2 \mathrm{x}_{\mathrm{n}}$ is $4 \overline{\mathrm{x}}$.

A number $n$ is randomly selected from the set $\{1,2,3, \ldots ., 1000\}$. The probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer is

The frequency distribution of daily working expenditure of families in a locality is as follows:  If the mode of the distribution is $₹ 140$, then the value of $b$ is

If six students, including two particular students $A$ and $B$, stand in a row, then the probability that $A$ and $B$ are separated with one student in between them is

There are two balls in an urn. Each ball can be either white or black. If a white ball is put into the urn and there after a ball is drawn at random from the urn, then the probability that it is white is

Statement 1: The variance of first $n$ odd natural numbers is $\frac{n^2-1}{3}$ Statement 2: The sum of first $\mathrm{n}$ odd natural number is $n^2$ and the sum of square of first $n$ odd natural numbers is $\frac{n\left(4 n^2+1\right)}{3}$.

Three numbers are chosen at random without replacement from $\{1,2,3, \ldots . .8\}$. The probability that their minimum is $3$ , given that their maximum is $6$ , is

If the mean of $4,7,2,8,6$ and a is 7 , then the mean deviation from the median of these observations is

The median of 100 observations grouped in classes of equal width is 25 . If the median class interval is 20-30 and the number of observations less than 20 is 45 , then the frequency of median class is