Random Variable
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: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
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Random Variable
- 1. Random Variable Bipul Kumar Sarker Lecturer BBA Professional Habibullah Bahar University College Chapter-02
- 2. Definitions and basic concepts: Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC The following definitions and terms are used in studying the theory of probability distribution. Random variable: A variable whose value is a number determined by the outcome of a random experiment is called a random variable. There are two types of random variable in statistics such as follow: ο Discrete Random Variable ο Continuous Random Variable
- 3. Discrete random variable: Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC Discrete random variable: If a random variable takes only a finite or a countable number of values, it is called a discrete random variable. Example: When 3 coins are tossed, the number of heads obtained is the random variable X assumes the values 0,1,2,3 which form a countable set. Such a variable is a discrete random variable.
- 4. Continuous random variable: Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC A random variable X which can take any value between certain interval is called a continuous random variable. Example: The height of students in a particular class lies between 4 feet to 6 feet. We write this as X = {x|4 ο£ο x ο£ο 6} The maximum life of electric bulbs is 2000 hours. For this the continuous random variable will be X = {x | 0 ο£ο x ο£ο 2000}
- 5. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC 2.2 Probability mass function: Let X be a discrete random variable which assumes the values π₯1, π₯2, β¦ β¦ β¦ π₯ π with each of these values, we associate a number called the probability ππ = π(π = π₯1), i = 1,2,3β¦n. This is called probability of π₯π satisfying the following conditions. i. Pi β₯ 0 πππ πππ π, π. π ππ β² π πππ πππ πππ β πππππ‘ππ£π. ii. ππ = π1 + π2 + β― β¦ β¦ β¦ β¦ β¦ β¦ . . +ππ = 1 π. π π‘ππ π‘ππ‘ππ ππππππππππ‘π¦ ππ πππ This function Pi or π(ππ) is called the probability mass function of the discrete random variable X. The set of all possible ordered pairs (x, p(x)) is called the probability distribution of the random variable X.
- 6. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC Note: The concept of probability distribution is similar to that of frequency distribution. Just as frequency distribution tells us how the total frequency is distributed among different values (or classes) of the variable, a probability distribution tells us how total probability 1 is distributed among the various values which the random variable can take. It is usually represented in a tabular form given below: X π₯1 π₯2 π₯3 β¦β¦β¦β¦.. π₯ π P(X = x) π(π₯1) π(π₯2) π(π₯3) β¦β¦β¦β¦.. π(π₯ π)
- 7. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC 2.2.1 Discrete probability distribution: If a random variable is discrete in general, its distribution will also be discrete. For a discrete random variable X, the distribution function or cumulative distribution is given by F(x) and is written as F(x) = P(X ο£ο x) ; - ο₯ο < x < ο₯ Thus in a discrete distribution function, there are a countable number of points π₯1, π₯2, β¦ β¦ β¦ and their probabilities Pi such that πΉ π₯π = Pi π₯ π< π₯ ; π = 1,2,3, β¦ β¦ β¦ β¦ . π
- 8. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC 2.2.2 Probability density function (pdf): A function f is said to be the probability density function of a continuous random variable X, if it satisfies the following properties. i. π π ο³ο 0 ; -ο₯ο < x < ο₯ ii. π π = π π π π = π β ββ
- 9. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC Remark: In case of a discrete random variable, the probability at a point ie P(x = a) is not zero for some fixed β aβ However in case of continuous random variables the probability at a point is always zero i.e π· πΏ = π = π π π π = π π π Hence P( a ο£ο X ο£ο b) = P(a < X < b) = P(a ο£ο X < b) = P(a < X ο£ο b) The probability that x lies in the interval (a,b) is given by π· π < πΏ < π = π(π) π π π π
- 10. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC Distribution function for continuous random variable: If X is a continuous random variable with p.d.f f(x), then the distribution function is given by i. π πΏ = π π π ββ π π = π· πΏ β€ π ; ββ < π < β i. π π β π π = π π π π π π = π· π < πΏ < π
- 11. Definitions and basic concepts: Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC
- 12. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC
- 13. Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC
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