The number of cars in a household is given below.

No. of cars	0	1	2	3	4
No. of Household	30	320	380	190	80

Estimate the probability mass function. Verify p(x_i ) is a probability mass function.

A random variable X has the following probability function

Values of X	0	1	2	3	4	5	6	7
p(x)	0	a	2a	2a	3a	a2	2a2	7a2+a

(i) Find a, Evaluate
(ii) P(X < 3),
(iii) P(X > 2) and
(iv) P(2 < X \(\leq\) 5).

\(\text { If } \ p(x) \ = \begin{cases}\frac{x}{20}, & x=0,1,2,3,4,5 \\ 0, & \text { otherwise }\end{cases}\)
Find
(i) P(X<3) and 
(ii) P(2\(\leq\)4)

Two unbiased dice are thrown simultaneously and sum of the upturned faces considered as random variable. Construct a probability mass function.

A coin is tossed thrice. Let X be the number of observed heads. Find the cumulative distribution function of X.

Construct the distribution function for the discrete random variable X whose probability distribution is given below. Also draw a graph of p(x) and F(x).

X = x	1	2	3	4	5	6	7
P(x)	0.10	0.12	0.20	0.30	0.15	0.08	0.05

A continuous random variable X has the following p.d.f f(x) = ax, 0\(\le\)x\(\le\)1
Determine the constant a and also find P\(\\ \left[ X\le \frac { 1 }{ 2 } \right] \)

A continuous random variable X has p.d.f
f(x) = 5x⁴, 0\(\le\)x\(\le\)1 
Find a₁ and a₂ such that
i) P[X\(\le\)a₁] = P[X>a₁]   
ii) P[X>a₂] = 0.05

Suppose, the life in hours of a radio tube has the following p.d.f
\(f(x)=\left\{\begin{array}{l} \frac{100}{x^{2}}, \text { when } x \geq 100 \\ 0, \text { when } x<100 \end{array}\right.\)
Find the distribution function.

The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the probability density function f(x) is given  by
\(f(x)=\left\{\begin{array}{l} Ax,for \ 0≤x10 \\ A(20−x),for \ 10 ≤x< 20 \\ 0,\quad \quad \quad otherwise \end{array}\right.\)
(a) Find the value of A.
(b) What is the probability that the number of pounds of bread that will be sold tomorrow is
(i) More than 10 pounds,
(ii) Less than 10 pounds, and
(iii) Between 5 and 15 pounds?