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Lesson Posted on 11 Jan Tuition/BTech Tuition/Engineering Mathematics (M1) Tuition/BTech Tuition/Advanced Mathematics (M2) Tuition/BSc Tuition/BSc Mathematics

Probability: Random Variable, Discrete Probability Distribution And Expectation

Sreedevi S.

Current Profession - Assistant Professor in Mathematics, University, Punjab Total Experience - 14 years Part...

An experiment whose outcome cannot be predicted is called random experiment. The set of all outcomes of a random experiment is called a sample space and is denoted by S. A sample Space is discrete if it has finitely many or a count ably infinite number of elements. A sample space is continuous it contains... read more

An experiment whose outcome cannot be predicted is called random experiment. The set of all outcomes of a random experiment is called a sample space and is denoted by S. A sample Space is discrete if it has finitely many or a count ably infinite number of elements. A sample space is continuous it contains uncountable number of elements. Any subset of a sample space is called event. is called an impossible event and whole space is called a sure event.

i. Random variable: A real valued function, defined over the sample space of a random experiment is called the random variable associated to that random experiment. That is the values of a random variable correspond to the outcomes of a random experiment.

Example: In the case of tossing three coins, the outcomes can be described as getting ‘0’ head, 2 heads and 3 heads. Let us consider a variable ‘X’ which takes values 0, 1, 2, 3. The value of X correspond to the outcomes 0 Head, 1 Head, 2 Heads and 3 Heads. Then X can be considered to be a random variable associated to the Random experiment of tossing three coins.

ii. Discrete and Continuous Random Variables: A random variable may be discrete or continuous. A random variable is said to be discrete if it assumes only specified values in an interval. When X takes values 1, 2, 3, 4, 5, 6 it is a discrete variable.

iii. A random variable is said to be continuous if it can assume any value in a given interval. When X takes any value in a given interval (a, b), it is a continuous variable in that interval.

iv. Distribution (Probability distribution) (Probability function) (Density function): Let ‘X’ be a random variable assuming values x_{1}, x_{2}, x_{3}….. Let ‘x’ stand for any one of x_{1}, x_{2}, x_{3 }…… Then probability that the random variable ‘X’ takes the value x is defined as probability function of ‘X’ and is denoted by f(x) or P(x). Therefore P(x) = P(X = x) where X is the random variable and ‘x’ stands for values of X, is the probability function of x.

When X takes values x_{1}, x_{2} … we have the corresponding probabilities P(x_{1}), P(x_{2})…such that P(x_{1}) + P(x_{2}) +………..= 1

i.e. ∑ P(x) = 1 and all P(x) >0

Example: In tossing two coins the random variable representing the number of heads takes the values x = 0, 1, 2…….

P (no head) = 1/4 Therefore P(x=0) = 1/4

P (one head) = 2/4 Therefore P(x=1) = 2/4

P (two heads) = 1/4 Therefore P(x=2) = 1/4

v. Properties of Discrete Probability Distribution:

Let P(x) be the probability density function, then,

P(x) > 0 for all values of x

- ∑ P(x) = 1

vi. Expectation of X (Expected value of X): Let the random variable X assume that the values of x_{1}, x_{2 … }with corresponding probabilities P(x_{1}), P(x_{2})… Then the expected value of the random variable X denoted by E (x) is given by the formula.

E(x) = x_{1} p (x_{1}) + x_{2} p (x_{2}) + ……….. I.e. E(x) = ∑ x p (x)

Example: When a die is thrown, the random variable X (number on the die takes the values 1, 2, 3, 4 ,5, 6 with corresponding probabilities 1/6, 1/6, 1/6, 1/6, 1/6, 1/6. Then

E(x) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 21/6

Variance of X: Variance of a random variable X, whose expectation denoted by E (x), is defined as

V (x) = E (x^{2}) – [E(x)]^{ 2}

^{ }

vii. Discrete Distribution function: Let X is a random variable and ‘x’ be any value of it. Then P[X < x] denoted by F (x) is called distribution function.

viii. Properties of distribution function:

- F (x) is a non decreasing function
- F (x) > 0
- F (∞) = 1.

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Comments Asked on 18/10/2017 Tuition/BSc Tuition/BSc Mathematics

How to find the rational roots of a quadratic equation?

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Asked on 18/09/2017 Tuition/BSc Tuition/BSc Mathematics

X, is assumed to be a random variable by placing letters of the word YACHT in a hat. If these letters... read more

X, is assumed to be a random variable by placing letters of the word ?YACHT? in a hat. If these letters are withdrawn then it gets replaced. A value of 1 is given when a vowel is retrieved, and a value of 2 is given when a consonant is retrieved while a value of 3 is given to all other retrieved letters. 1. Write the sample space on which X is defined? 2. Construct a probability distribution of X. read less

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Asked on 16/09/2017 Tuition/BSc Tuition/BSc Mathematics

Probability Question: X, is assumed to be a random variable by placing letters of the word YACHT in... read more

Probability Question: X, is assumed to be a random variable by placing letters of the word ?YACHT? in a hat. If these letters are withdrawn then it gets replaced. A value of 1 is given when a vowel is retrieved, and a value of 2 is given when a consonant is retrieved while a value of 3 is given to all other retrieved letters. Write down the sample space on which X variable is defined.Can we write the sample space as given below: Since Y-2, A-1, C-2, H-2, T-2 Sample space = {1 and 2}. read less

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Asked on 21/08/2017 Tuition/BSc Tuition/BSc Mathematics

Zero raised to the power zero is ........?? Prove.

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Answered on 16/12/2016 Tuition/BSc Tuition Tuition/BSc Tuition/BSc Mathematics

Margshree Classes

The locus of tangent drawn from a fixed point to a sphere( or conicoid ) is a cone called enveloping cone or tangent from the point to the sphere.

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Answers 15 Comments Looking for BSc Tuition

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Answered on 07/12/2016 Tuition/BSc Tuition Tuition/BSc Tuition/BSc Mathematics

Sarvajeet Kumar

An Experienced Trainer

Computer (desktop or laptop) and internet connection.

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Answers 14 Comments Answered on 29/09/2016 Tuition/BSc Tuition Tuition/BSc Tuition/BSc Mathematics

Shruti

Technical tutor

The simplest answer can be a communicating device which can support video chat.. like computer, laptop, smartphone, netbook etc

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Answers 9 Comments Lesson Posted on 02/09/2016 Tuition/BSc Tuition Tuition/BSc Tuition/BSc Mathematics IT Courses/Data Science

Studying mathematics and related subjects

Rohan

learning mathematical concepts requires two preconditions - that you understand and write rigorous proofs for even simple concepts and that you understand it intuitively. If either you didnt develop an intuition or you couldnt prove what you say is true, you should stop and try to work hard till you... read more

learning mathematical concepts requires two preconditions - that you understand and write rigorous proofs for even simple concepts and that you understand it intuitively.

If either you didnt develop an intuition or you couldnt prove what you say is true, you should stop and try to work hard till you get both the proofs and the intuition.

over a certain period of time, this practice will deepen your intuition overall and you would be able to penetrate though the mathematical concepts if continued this for a long time ..........

Also listen to classical music - Indian classical in particular

ok fellas....keep up the good work.

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Lesson Posted on 31/08/2016 Tuition/BSc Tuition/BSc Physics Tuition/BSc Tuition/BSc Chemistry Tuition/BSc Tuition/BSc Mathematics

Cultivate the habit of writing whatever you have studied.

Sivakumar

I was a Trainer in a Pharmaceutical Company, I was training medical representatives and other sales oriented...

Reading is a must and very important but periodically write down what you have learned in your own words. This is really very helpful when you are preparing for the exams. Try to derive equations (Physics, Chemistry) solve mathematical problems or anything you feel you are challenging. This way you will... read more

Reading is a must and very important but periodically write down what you have learned in your own words. This is really very helpful when you are preparing for the exams. Try to derive equations (Physics, Chemistry) solve mathematical problems or anything you feel you are challenging. This way you will prepare indirectly for your exams. Imagine you are solving an equation for the first time in the examination hall. Any error will stop you at that point itself and you grow nervous if you are not able to solve it. You will lose time and you will not have adequate time to complete the question paper. So take it seriously. Write down periodically whatever you have studied. Solve equations or problems several times till you are confident that you can do easily. Take several mock tests before the exams. You can get previous year question papers over the internet or they are available with the book sellers. I once again reiterate that writing is a very good practice to score good marks in the exams.

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