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.

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.

Construct cumulative distribution function for the given probability distribution.

X	0	1	2	3
P(X = x)	0.3	0.2	0.4	0.1

The discrete random variable X has the probability function

X	1	2	3	4
P(X=x)	k	2k	3k	4k

Show that k = 0.1.

Two coins are tossed simultaneously. Getting a head is termed as success. Find the probability distribution of the number of successes.

Define random variable.

Explain what are the types of random variable?

Define discrete random variable.

What do you understand by continuous random variable?

Describe what is meant by a random variable.

Distinguish between discrete and continuous random variable.

Explain the distribution function of a random variable.

Six men and five women apply for an executive position in a small company. Two of the applicants are selected for an interview. Let X denote the number of women in the interview pool. We have found the probability mass function of X.

X = x	0	1	2
P(x)	\(\frac{2}{11}\)	\(\frac{5}{11}\)	\(\frac{4}{11}\)

How many women do you expect in the interview pool?

The following information is the probability distribution of successes.

No. of Successes	0	1	2
Probability	\(\frac{6}{11}\)	\(\frac{9}{22}\)	\(\frac{1}{22}\)

Determine the expected number of success.

Find the expected value for the random variable of an unbiased die

Let X be a random variable defining number of students getting A grade. Find the expected value of X from the given table

X = x	0	1	2	3
P(X=x)	0.2	0.1	0.4	0.3

Let X be a continuous random variable with probability density function
\({ f }_{ x }(x)=\begin{cases} \begin{matrix} 2x, & 0\le x\le 1 \end{matrix} \\ \begin{matrix} 0, & otherwise \end{matrix} \end{cases}\)
Find the expected value of X.

In an investment, a man can make a profit of Rs. 5,000 with a probability of 0.62 or a loss of Rs. 8,000 with a probability of 0.38. Find the expected gain.

What are the properties of Mathematical expectation?

What do you understand by Mathematical expectation?

How do you define variance in terms of Mathematical expectation?

Define Mathematical expectation in terms of discrete random variable.

State the definition of Mathematical expectation using continuous random variable.

Let X be a random variable and Y = 2X + 1. What is the variance of Y if variance of X is 5 ?

Prove that if E(X) = 0, then V(X) = E(X²)

What is the expected value of a game that works as follows: I flip a coin and, if tails pay you Rs. 2; if heads pay you Rs. 1. In either case I also pay you Rs. 50.

Prove that,  V(aX) = a²V(X)

The time to failure in thousands of hours of an important piece of electronic equipment used in a manufactured DVD player has the density function
\(f(x)= \begin{cases}2 e^{-2 x}, & x>0 \\ 0,& \text { otherwise }\end{cases}\)
Find the expected life of this piece of equipment.

Prove that, V(X+b) = V(X)