Which of the following statements are correct?

(A) The mean of a binomial distribution is always greater than the variance.

(B) The mean of a Poisson distribution is always greater than the variance.

(C) If $f(x)$ is the probability density function of a random variable $X$, then $P(a \leq X \leq b) = \int_a^b f(x)dx$.

(D) If $X$ is a random variable and $a,b$ are real numbers, then $Var(aX+b)=a^2Var(X)$.

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Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	In a binomial distribution, if $n = 16$, $p = 0.75$, then the mean is	I.	$4$
B.	In a binomial distribution, if $n = 25$, $p = \dfrac{1}{5}$, then variance is	II.	$32$
C.	In a binomial distribution, the mean is 8 and the variance is 6, then the number of trials is	III.	$18$
D.	If the mean and variance of a binomial distribution are 6 and 4 respectively, then the number of trials is	IV.	$12$

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The probability distribution of a random variable $X$ is given as $P(X=x)=\begin{cases}kx^3 & \text{for } x=1,2,3\\ 3kx & \text{for } x=4,5\\ 0 & \text{otherwise}\end{cases}$, where $k$ is a constant.

The value of $P(X\ge4)$ is

If $X$ is a discrete random variable which assumes values $x_1, x_2, x_3, ..., x_n$ with the respective probabilities $p_1, p_2, ....., p_n$, then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\displaystyle\sum_{i=1}^{n} p_i =$	I.	$E(X^2)$
B.	$\displaystyle\sum_{i=1}^{n} p_i x_i =$	II.	$Var(X)$
C.	$\displaystyle\sum_{i=1}^{n} p_i x_i^2 =$	III.	$1$
D.	$E(X^2) - \{E(X)\}^2 =$	IV.	$E(X)$

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The random variable X has a probability distribution $P(X)$ of the following form, where $k$ is some real number.

$P(X=x) = \begin{cases}k, & \text{if } x=0\\\dfrac{k}{2}, & \text{if } x=1 \text{ or } 2\\0, & \text{otherwise}\end{cases}$

On the basis of the above information, match List-I with List-II

LIST-I		LIST-II
A. Value of $k$		I. $\dfrac{1}{4}$
B. $P(X<2)$		II. 0
C. $P(X>1)$		III. $\dfrac{1}{2}$
D. $P(X>2)$		IV. $\dfrac{3}{4}$

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Let X denote the number of scores in a test. If X is normally distributed with mean 100 and standard deviation 15, then the probability that X does not exceed 130 is,

(Given $P(0 \le Z \le 2)=0.4772$, where $Z$ be the standard normal variate)

If a coin is tossed 8 times and getting a head is considered as success, then which of the following statements are correct ?

(A) The probability of getting 4 heads is $\dfrac{35}{128}$

(B) The mean of the number of successes is 4.

(C) The variance of the number of successes is 2.

(D) The probability of getting 6 heads is $\dfrac{9}{64}$.

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