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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 +2 Tuition/BA Tuition/Bio-Mathematics Tuition/BCom Tuition/Business Mathematics and Statistics less

Probability: Random Variable, Discrete Probability Distribution And Expectation

Sreedevi S.

Current Profession - Assistant Professor in Mathematics, Chitkara University, Punjab Total Experience...

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 x1, x2, x3….. Let ‘x’ stand for any one of x1, x2, x3 …… 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 x1, x2 … we have the corresponding probabilities P(x1), P(x2)…such that P(x1) + P(x2) +………..= 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 x1, x2 … with corresponding probabilities P(x1), P(x2)… Then the expected value of the random variable X denoted by E (x) is given by the formula.

E(x) = x1 p (x1) + x2 p (x2) + ……….. 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 (x2) – [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:

  1. F (x) is a non decreasing function
  2. F (x) > 0
  3. F (∞) = 1.
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X, is assumed to be a random variable by placing letters of the word YACHT in a hat. If these letters... read more
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Probability Question: X, is assumed to be a random variable by placing letters of the word YACHT in... read more
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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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Computer (desktop or laptop) and internet connection.
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The simplest answer can be a communicating device which can support video chat.. like computer, laptop, smartphone, netbook etc
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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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