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Let X<sub>1</sub>, X<sub>2</sub> be two independent normal random variables with means μ<sub>1</sub>, μ<sub>2</sub> and standard deviations σ<sub>1</sub>, σ<sub>2</sub> respectively. Consider Y = X<sub>1</sub> - X<sub>2</sub>; μ<sub>1</sub> = μ<sub>2</sub> = 1, σ<sub>1</sub> = 1, σ<sub>2</sub> = 2, Then,

Correct Answer: Y is normally distributed with mean 0 and variance 5

Which of the following statements relating to correlation and reegression are true?
1. The coefficient of correlation is independent of change of origin and scale
2. The coefficient of correlation between the two variables is the arithmetic average of the two regression coefficients
3. The probable error of the coefficient correlation is 0.6745 times of its standard error
4. Coefficient of correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line
Select the correct answer:

What is linear_congruential_engine? for some w using Mersenne Twister algorithm b) Pseudo-random number engine that generates random unsigned integers c) Pseudo-random number engine that generates random unsigned integers in the range for some w using lagged Fibonacci generator d) Pseudo-random number engine that generates random signed integers in the range for some w using Mersenne Twister algorithm

Let X1 and X2 be two independent exponentially distributed random variables with means 0.5 and 0.25 respectively. Then Y = min (X1, X2) is

Let X be a random variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown. If P(X ≤ -1) = P(Y ≥ 2) the standard deviation of Y is

Statement I The product of coefficient of correlation, standard deviation of variable 'X' and Standard deviation of variable 'Y' gives the measure of covariance between X and Y variables.
Statement II The product of coefficient of correlation between X and Y variables and the ratio between standard deviation of X variable to standard deviation of Y variable measures the slope of the regression line of X on Y variable.

Assertion (A): When there is an evidence of a linear relationship between two variables, it may not always mean an independent-dependent relationship between the two variables.
Reason (R): The casual relationship between the two variables may not imply a reasonable theoretical relationship between the two.

Suppose X and Y are random variables with ?x = 32, ?x = 5, ?y = 44, and ?y = 12. Given that X and Y are independent, what are the mean and standard deviation of the random variable X + Y?

X and Y are two independent random variables with variances 1 and 2, respectively. Let Z = X - Y. The variance of Z is

The standard normal cumulative probability function (probability from $$ - \infty $$  to xn) can be approximated as $${\text{F}}\left( {{{\text{x}}_{\text{n}}}} \right) = \frac{1}{{1 + {\text{exp}}\left( { - 1.7255{{\text{x}}_{\text{n}}}{{\left| {{{\text{x}}_{\text{n}}}} \right|}^{0.12}}} \right)}}$$       where xn = standard normal deviate. If mean and standard deviation of annual precipitation are 102 cm and 27 cm respectively, the probability that the annual precipitation will be between 90 cm and 102cm is

A,B,C,D,E,G and I are seven friends who study in three different standards namely 5th, 6th and 7th such that not less than two friends study in the same standard. Each friend also has a different favorite subject namely History, Civics, English, Bangla, Science , Maths , and Economics but not necessarily in the same order. A likes Maths and studies in the 5th standard with only one other friend who likes Bangla. I studies with tow other friends . Both the friends who study with I like languages (Here languages include only Science , English and Bangla ) . D studies in the 6th standard with only one person and does not like Civics . E studies with only one friend . The one who likes history does not study in 5th or 6th standard . E does not like languages. C does not like English , Science or Civics. Which combination represents E's favorite subject and the standard in which he studies?