A discrete random variable X has the following probability distribution.

x	1	2	3	4	5	6	7
P(X)	c	2c	2c	3c	c2	2c2	7c2+c

Find the value of e. Also, find the mean of the distribution.

The probability distribution of a random variation X is given below.

X	0	1	2	3	4
P(X)	0.1	0.25	0.3	0.2	0.15

Find
(i) V(X)
ii) V\((\frac{X}{2})\)

The probability distribution of the discrete random variables X and Y are given below

X	0	1	2	3
P(X)	\(\frac{1}{5}\)	\(\frac{2}{5}\)	\(\frac{1}{5}\)	\(\frac{1}{5}\)

Y	0	1	2	3
P(Y)	\(\frac{1}{5}\)	\(\frac{3}{10}\)	\(\frac{2}{5}\)	\(\frac{1}{10}\)

Prove that E(Y²) = 2E(X).

The random variable X tan take only the values 0,1,2. Given that P(X = 0) = P(X = 1) = P and E(X²) = E(X), find the value of p.

The probability distribution of a random variable X is

X	1	2	4	2A	3A	5A
P(X)	\(\frac{1}{2}\)	\(\frac{1}{5}\)	\(\frac{3}{25}\)	\(\frac{1}{10}\)	\(\frac{1}{25}\)	\(\frac{1}{25}\)

Calculate
(i) A if E(X) = 2.94
(ii) V(X)

Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the value X has the following form, where k is some unknown constant \(p(X=x)= \begin{cases}0.1 & \text { if } x=0 \\ k x & \text { if } x=1 \text { or } 2 \\ k(5-x) & \text { if } x=3 \text { or } 4 \\ 0 & \text { otherwise }\end{cases}\)
(i) Find the value of k 
(ii) What is the probability that you study atleast 2 hours? 
(iii) Exactly 2 hours
(iv) At most 2 hours

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of two numbers obtained. Find E(X) and var (X).

Find the probability distribution of the number ofsuccess in 2 tosses of a die, where a success in defined as "getting a number greater than 4". Also find the mean and variance of the distribution

A fair coin is tossed until a head or five tails occur. If X denotes the number of tosses of the coin, find mean of X

A random variable X has the following probability mass function 

x	0	1	2	3	4	5	6
P(X=x)	k	3k	5k	7k	9k	11k	13k

(i) Find k
(ii) Evaluate \(\mathbf{P}(x<4), P(x \geq 5), \quad \mathrm{P}(3<x \leq 6)\)
(iii) What is the smallest value of x for which \(\mathrm{P}(X \leq x)>\frac{1}{2}\)

A continuous random variable X has p.d.f \(f(x)=3 x^{2}, \ 0 \leq x \leq 1\). Find a and b such that 
(i) \(\mathrm{P}(x \leq \mathrm{a})=\mathrm{P}(x>\mathrm{a})\)
(ii) \( \mathrm{P}(x>\mathrm{b})=0.05\)

If the probability density function of a random variable is given by \(f(x)= \begin{cases}k\left(1-x^{2}\right) & 0<x<1 \\ 0 & \text { elsewhere }\end{cases}\)
(i) Find k
(ii) The distribution function of the random variable

For the p.d.f \(f(x)=\left\{\begin{array}{lc} \operatorname{Cx}(1-x)^{3}, & 0<x<1 \\ 0, & \text { elsewhere } \end{array}\right.\) 
(i) the constant C
\(\text {(ii) } \mathrm{P}\left(x<\frac{1}{2}\right)\)

A continuous random variable has the following p.d.f, \(f(x)= \begin{cases}k x^{2}, & 0 \leq x \leq 10 \\ 0, & \text { otherwise }\end{cases}\) find k and evaluate
\((i)\ \mathrm{P}(0.2 \leq x \leq 0.5) \)
(ii) \( \mathrm{P}(x \leq 3)\)

Let X be a continuous random variable with \(\text { p.d.f } f(x)= \begin{cases}a x & 0 \leq x \leq 1 \\ a & 1 \leq x \leq 2 \\ -a x+3 a & 2 \leq x \leq 3 \\ 0 & \text { otherwise }\end{cases}\)
(i) Determine the constant a
(ii) Compute \(\mathrm{P}(x \leq 1.5)\)