The probability distribution function of a normal variate with mean $\mu$ and variance $\sigma^2$ is given by: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$, $-\infty < x < \infty$, $-\infty < \mu < \infty$, $\sigma > 0$ If $y = f(x)$ be the normal probability curve, then which of the following is correct? (A) The normal curve is symmetrical about the line $x = \mu$. (B) Mean, median and mode of the distribution coincide. (C) Y- axis is an asymptote to the normal curve. (D) If x increases numerically, $f(x)$ decreases rapidly. Choose the correct answer from the options given below:

A random variable $X$ has the following probability distribution:

X	0	1	2	3	4
P(X)	0	$\dfrac{k}{2}$	$\dfrac{3k}{2}$	$\dfrac{k}{3}$	$2k$

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	Value of $k$	I.	$\dfrac{7}{13}$
B.	Value of $P(X \leq 3)$	II.	$\dfrac{3}{26}$
C.	Value of $P(X > 1)$	III.	$\dfrac{23}{26}$
D.	Value of $P(X < 2)$	IV.	$\dfrac{3}{13}$

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If the events A and B are independent, then which of the following statements are true?

(A) P(A'B) = [1-P(A)] P(B)

(B) A and B are mutually exclusive

(C) P(A) = P(B)

(D) P(A'B') = [1-P(A)] [1-P(B)]

Choose the correct answer from the options given below:

Let A and B are two events associated to a random experiment, then

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. $P(A \cap B) + P(A \cap \overline{B})$	I. $P(A \cup B)$
B. $P(A \cap B) + P(\overline{A} \cap B)$	II. $P(A) + P(B)$
C. $P(A \cap B) + P(\overline{A} \cap B) + P(A \cap \overline{B})$	III. $P(A)$
D. $P(A \cup B) + P(A \cap B)$	IV. $P(B)$

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Which of the following statements are NOT correct about Standard Normal Distribution?

(A) The probability curve of the Standard Normal Distribution is a bell-shaped curve.

(B) The Standard Normal variate (Z) score describes the position of each data point in terms of its distance from the mean, when measured in standard deviation units.

(C) The Z-score is negative if the data point lies above the mean, and positive if it lies below the mean.

(D) There is a 95.45 % probability of randomly selecting a score between $\mu - \sigma$ and $\mu + \sigma$, when $\sigma$ is standard deviation and $\mu$ is mean.

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Let A and B be independent events such that $P(A) = \frac{3}{7}$ and $P(B) = \frac{1}{7}$, Then

Match List-I with List-II

List-I	List-II
(A) P(A and B)	(I) $\frac{24}{49}$
(B) P(A or B)	(II) $\frac{18}{49}$
(C) P(A and not B)	(III) $\frac{25}{49}$
(D) P(neither A nor B)	(IV) $\frac{3}{49}$

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Let X be a random variable. Let E (X) and Var (X) denote the mean and the variance of X respectively. Then match List-I with List-II

List-I	List-II
(A) If Var (X) = $a$, then Var (2X + 3) is	(I) 11$a$
(B) If E (X) = $a$, then E (2X) is	(II) 6$a$
(C) If Var (X) = $a$, then Var(3X - $a$) + Var ($\sqrt{2}x + \beta$) is	(III) 4$a$
(D) If E (X) = $\frac{5a}{12}$, then E (12X + $a$) is	(IV) 2$a$

Choose the correct answer from the options given below: