Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board chapter 6 - Random Variable and Mathematical expectation [Latest edition]

Construct cumulative distribution function for the given probability distribution.

Let X be a discrete random variable with the following p.m.f
`"P"(x) = {{:(0.3,  "for"  x = 3),(0.2,  "for"  x = 5),(0.3,  "for"  x = 8),(0.2,  "for"  x = 10),(0,  "otherwise"):}`
Find and plot the c.d.f. of X.

The discrete random variable X has the following probability function.
P(X = x) = `{{:("k"x,  x = 2","  4","  6),("k"(x - 2),  x = 8),(0,  "otherwise"):}`
where k is a constant. Show that k = `1/18`

The discrete random variable X has the probability function

Show that k = 0 1

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

The discrete random variable X has the probability function.

Find k

The discrete random variable X has the probability function.

Evaluate p(x < 6), p(x ≥ 6) and p(0 < x < 5)

The discrete random variable X has the probability function.

If P(X ≤ x) > `1/2`, then find the minimum value of x.

The distribution of a continuous random variable X in range (– 3, 3) is given by p.d.f.
f(x) = `{{:(1/16(3 + x)^2",", - 3 ≤ x ≤ - 1),(1/16(6 - 2x^2)",", - 1 ≤ x ≤ 1),(1/16(3 - x)^2",", 1 ≤ x ≤ 3):}`
Verify that the area under the curve is unity.

A continuous random variable X has the following distribution function
F(x) = `{{:(0",",  "if"  x ≤ 1),("k"(x - 1)^4",",  "if"  1 < x ≤ 3),(1",",  "if"  x > 3):}`
Find k

A continuous random variable X has the following distribution function
F(x) = `{{:(0",",  "if"  x ≤ 1),("k"(x - 1)^4",",  "if"  1 < x ≤ 3),(1",",  "if"  x > 3):}`
Find the Probability density function

The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as 
f(x) = `{{:("Ae"^((-x)/5)",",  "for"  x ≥ 0),(0",",  "otherwise"):}`
Find the value of A that makes f(x) a p.d.f.

The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as 
f(x) = `{{:("Ae"^((-x)/5)",",  "for"  x ≥ 0),(0",",  "otherwise"):}`
What is the probability that the number of minutes that person will talk over the phone is (i) more than 10 minutes, (ii) less than 5 minutes and (iii) between 5 and 10 minutes

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function

F(x) = `{{:(0",",  "for"  x ≤ 0),(x/2",",  "for"  0 ≤ x < 1),(1/2",",  "for" ≤ x < 2),(x/4",",  "for"  2 ≤ x < 4),(1",",  "for"  x ≥ 4):}` 
Is the distribution function continuous? If so, give its probability density function?

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function

F(x) = `{{:(0",",  "for"  x ≤ 0),(x/2",",  "for"  0 ≤ x < 1),(1/2",",  "for" ≤ x < 2),(x/4",",  "for"  2 ≤ x < 4),(1",",  "for"  x ≥ 4):}` 
What is the probability that a person will have to wait (i) more than 3 minutes, (ii) less than 3 minutes and (iii) between 1 and 3 minutes?

Define random variable

Explain what are the types of random variable?

Define dicrete random Variable

What do you understand by continuous random variable?

Describe what is meant by a random variable

Distinguish between discrete and continuous random variables.

Explain the distribution function of a random variable

Explain the terms probability Mass function

Explain the terms probability density function

Explain the terms probability distribution function

What are the properties of discrete random variable

What are the properties of continuous random variable?

State the properties of distribution function.

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:

The following table is describing about the probability mass function of the random variable X

Find the standard deviation of x.

Let X be a continuous random variable with probability density function
`"f"_x(x) = {{:(2x",", 0 ≤ x ≤ 1),(0",",  "otherwise"):}`
Find the expected value of X

Let X be a continuous random variable with probability density function
f(x) = `{{:(3/x^4",",  x ≥ 1),(0",",  "otherwise"):}`
Find the mean and variance of X

In investment, a man can make a profit of ₹ 5,000 with a probability of 0.62 or a loss of ₹ 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 defi ne 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

In a business venture a man can make a profit of ₹ 2,000 with a probability of 0.4 or have a loss of ₹ 1,000 with a probability of 0.6. What is his expected, variance and standard deviation of profit?

The number of miles an automobile tire lasts before it reaches a critical point in tread wear can be represented by a p.d.f.
f(x) = `{{:(1/30 "e"^(- x/30)",",  "for"  x > 0),(0",",  "for"  x ≤ 0):}`
Find the expected number of miles (in thousands) a tire would last until it reaches the critical tread wear point

A person tosses a coin and is to receive ₹ 4 for a head and is to pay ₹ 2 for a tail. Find the expectation and variance of his gains

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

Choose the correct alternative:

Value which is obtained by multiplying possible values of a random variable with a probability of occurrence and is equal to the weighted average is called

Choose the correct alternative:

Demand of products per day for three days are 21, 19, 22 units and their respective probabilities are 0.29, 0.40, 0.35. Profit per unit is 0.50 paisa then expected profits for three days are

Choose the correct alternative:

Probability which explains x is equal to or less than a particular value is classified as

Choose the correct alternative:

Given E(X) = 5 and E(Y) = – 2, then E(X – Y) is

Choose the correct alternative:

A variable that can assume any possible value between two points is called

Choose the correct alternative:

A formula or equation used to represent the probability distribution of a continuous random variable is called

Choose the correct alternative:

If X is a discrete random variable and p(x) is the probability of X, then the expected value of this random variable is equal to

Choose the correct alternative:

Which of the following is not possible in probability distribution?

Choose the correct alternative:

If c is a constant, then E(c) is

Choose the correct alternative:

A discrete probability distribution may be represented by

Choose the correct alternative:

A probability density function may be represented by

Choose the correct alternative:

If c is a constant in a continuous probability distribution, then p(x = c) is always equal to

Choose the correct alternative:

E[X – E(X)] is equal to

Choose the correct alternative:

E[X – E(X)]² is

Choose the correct alternative: 

If the random variable takes negative values, then the negative values will have

Choose the correct alternative: 

If we have f(x) = 2x, 0 ≤ x ≤ 1, then f(x) is a

Choose the correct alternative: 

`int_(-oo)^oo` f(x) dx is always equal to

Choose the correct alternative: 

A listing of all the outcomes of an experiment and the probability associated with each outcome is called

Choose the correct alternative: 

Which one is not an example of random experiment?

Choose the correct alternative: 

A set of numerical values assigned to a sample space is called

Choose the correct alternative: 

A variable which can assume finite or countably infinite number of values is known as

Choose the correct alternative: 

The probability function of a random variable is defined as

Then k is equal to

Choose the correct alternative: 

If p(x) = `1/10`, x = 10, then E(X) is

Choose the correct alternative: 

A discrete probability function p(x) is always

Choose the correct alternative: 

In a discrete probability distribution, the sum of all the probabilities is always equal to

Choose the correct alternative: 

An expected value of a random variable is equal to it’s

Choose the correct alternative: 

A discrete probability function p(x) is always non-negative and always lies between

Choose the correct alternative: 

The probability density function p(x) cannot exceed

Choose the correct alternative: 

The height of persons in a country is a random variable of the type

Choose the correct alternative: 

The distribution function F(x) is equal to

The probability function of a random variable X is given by
p(x) = `{{:(1/4",",  "for"  x = - 2),(1/4",",  "for"  x = 0),(1/2",",  "for"  x = 10),(0",",  "elsewhere"):}`
Evaluate the following probabilities
P(X ≤ 0)

The probability function of a random variable X is given by
p(x) = `{{:(1/4",",  "for"  x = - 2),(1/4",",  "for"  x = 0),(1/2",",  "for"  x = 10),(0",",  "elsewhere"):}`
Evaluate the following probabilities
P(X < 0)

The probability function of a random variable X is given by
p(x) = `{{:(1/4",",  "for"  x = - 2),(1/4",",  "for"  x = 0),(1/2",",  "for"  x = 10),(0",",  "elsewhere"):}`
Evaluate the following probabilities
P(|X| ≤ 2)

The probability function of a random variable X is given by
p(x) = `{{:(1/4",",  "for"  x = - 2),(1/4",",  "for"  x = 0),(1/2",",  "for"  x = 10),(0",",  "elsewhere"):}`
Evaluate the following probabilities
P(0 ≤ X ≤ 10)

Let X be a random variable with a cumulative distribution function.
F(x) = `{{:(0",",  "if"  x  < 0),(x/8",",  "if"  0 ≤ x ≤ 1),(1/4 + x/8",",  "if"  1 ≤ x ≤ 2),(3/4 + x/12",",  "if"  2 ≤ x < 3),(1",",  "for"  3 ≤ x):}`
Compute: (i) P(1 ≤ X ≤ 2) and (ii) P(X = 3)

Let X be a random variable with a cumulative distribution function.
F(x) = `{{:(0",",  "if"  x  < 0),(x/8",",  "if"  0 ≤ x ≤ 1),(1/4 + x/8",",  "if"  1 ≤ x ≤ 2),(3/4 + x/12",",  "if"  2 ≤ x < 3),(1",",  "for"  3 ≤ x):}`
Is X a discrete random variable? Justify your answer

The p.d.f. of X is defined as
f(x) = `{{:("k"",",  "for"  0 < x ≤ 4),(0",",  "otherwise"):}`
Find the value of k and also find P(2 ≤ X ≤ 4)

The probability distribution function of a discrete random variable X is
f(x) = `{{:(2k",",  x = 1),(3k",",  x = 3),(4k",", x = 5),(0",",  "otherwise"):}`
where k is some constant. Find k 

The probability distribution function of a discrete random variable X is
f(x) = `{{:(2k",",  x = 1),(3k",",  x = 3),(4k",", x = 5),(0",",  "otherwise"):}`
where k is some constant. Find P(X > 2) 

The probability density function of a continuous random variable X is
f(x) = `{{:("a" + "b"x^2",",  0 ≤ x ≤ 1),(0",",  "otherwise"):}`
where a and b are some constants. Find a and b if E(X) = `3/5`

The probability density function of a continuous random variable X is
f(x) = `{{:("a" + "b"x^2",",  0 ≤ x ≤ 1),(0",",  "otherwise"):}`
where a and b are some constants. Find Var(X)

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 ₹ 2; if heads pay you ₹ 1. In either case, I also pay you ₹ 0.50

Prove that V(aX) = a²V(X)

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

Consider a random variable X with p.d.f.
f(x) = `{(3x^2",",  "if"  0 < x < 1),(0",",  "otherwise"):}`
Find E(X) and V(3X – 2)

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) = `{{:(2"e"^(-2x)",",  x > 0),(0",",  "otherwise"):}`
Find the expected life of this piece of equipment