Learning Outcomes: ▪ Emphasize that probability distribution function is usually denoted by Construct the probability distribution table and probability distribution function. [Include: Graph sketching & mode.] (a) (b) Find the cumulative distribution function. [Include: Median] (c) Find the probabilities from At the end of the lesson, students should be able to (i) probability distribution table or function (ii) cumulative distribution function Find the expectation and variance of discrete random variables. [Include: Properties] (d) Discrete Random Variables (I) is a discrete random variable with the probability mass function, where if satisfy both the conditions below: a table, function or graph which list all the possible values of a discrete random variable with their corresponding probabilities. Discrete Probability Distribution The probability distribution function corresponding to a discrete random variable is known as probability mass function ( ) and denoted by . • •The mode of a discrete random variable is the value with the highest probability. Remarks: 8.1 Discrete Random Variables C08 -SM025 Page 1
Given 2 fair coins are tossed. (i) table (ii) function (iii) graph If represents the number of tail obtained, express the probability distribution of in the form of Solution: The possible outcomes are {HH, HT, TH, TT}. (i) Probability Distribution Table (ii) Probability Distribution Function (iii) Probability Distribution Graph Example 1 C08 -SM025 Page 2
If represent the number on the die, show that is a discrete random variable. (a) (b) Hence, find and A fair die is rolled. Solution: The possible outcomes are {1, 2, 3, 4, 5, 6}. (a) Since and for all , is a discrete random variable. [Shown] (b) Example 2 C08 -SM025 Page 3
A random variable has the following probability distribution function a) Determine the value of . Solution: b) Construct a probability distribution table. Solution: c) State the mode. Solution: d) Find (i) (ii) Solution: where is a constant. Example 3 C08 -SM025 Page 4
for , If is a discrete random variable with a p.m.f. as follows: then the cumulative distribution function of , is given by Remarks: • If exists, then If does not exists, then • To find median, of a discrete random variable, Example 4 a) Construct the cumulative distribution table. Hence, find the median. Solution: b) Sketch the graph of . Solution: The probability distribution of a discrete random variable is given in the table below. Discrete Cumulative Distribution C08 -SM025 Page 5
The random variable has the following probability distribution : Find the cumulative distribution function of . The cumulative distribution function of Solution: Example 5 C08 -SM025 Page 6
The cumulative distribution function of a discrete random variable is tabulated as follows. a) Find . Solution: b) Find median, Solution: If there is NO , then refer to started to be . c) Construct the probability distribution table. Solution: d) Find . Solution: Based on table in (c) Based on Example 6 *[Example 1 of Lecture 2 of 4 in PowerPoint] C08 -SM025 Page 7
Consider a discrete random variable with the p.m.f. . • also known as Expected Value of or Expectation of • denoted by or • given by ▪ ▪ ▪ ▪ Properties: [ are real number constants] Mean of denoted by or • • given by ▪ ▪ Properties: [ are real number constants] Variance of • denoted by • given by Standard Deviation of Mean & Variance (Discrete Random Variable) C08 -SM025 Page 8
The probability distribution of a discrete random variable is given as follows. Calculate Solution: Solution: Solution: Solution: Example 1 *[Example 2 in Lecture 2 of 5 in PowerPoint] C08 -SM025 Page 9
A discrete random variable has the following probability distribution: Solution: Solution: Solution: Solution: Example 2 *[Example 3 in Lecture 2 of 5 in PowerPoint] C08 -SM025 Page 10
The probability distribution of a discrete random variable X is given as follows: If , find the constant and . Solution: Example 3 *[Example 4 in Lecture 2 of 5 in PowerPoint] C08 -SM025 Page 11
A random variable has the following probability distribution: where is a constant. (a) Determine the value of Solution: Solution: Solution: Solution: Example 4 *[Exercise in Lecture 1 of 5 in PowerPoint] C08 -SM025 Page 12