EE 313 Probability for Engineers This lecture is

EE 313 Probability for Engineers This lecture is prepared on 4 th June 2019 by Asim Ul Haq, https: //www. engrasimulhaq. wordpress. com 1

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■ Course Instructor – Asim Ul Haq (Lab Engineer, EE Department, AUIC) – asim. ulhaq@abasynisb. edu. pk 3

Recall (Theorem of Total Probability) Chapter 2 ■ This is called Bayes If the events B , . . , B constitute a partition of the sample space S such that P(B ) ≠ 0 for i. Theorem = 1, 2, . . . , k, then for any event A of S 1 2 k i B 2 B 1 B 3 B 5 Bk A … B 4 4

Bayes’ Theorem (proof) Chapter 2 5

Bayes’ Theorem (proof) 6

Random Variable ■ A random variable is a function that associates a real number with each element in the sample space. ■ Capital letter is used to represent the random variable, ■ Small letter is used to represent the value ■ Example – Number of people visiting an ATM – Pressure of gas at different CNG stations – Tossing of coin? ? ? 7

Discrete and Continuous Random Variables ■ A Discrete Random Variable is the one that has a countable set of outcomes. ■ When a random variable takes on values on continuous scale, the variable is regarded as continuous random variable. 8

Example of Discrete Random Variables ■ Number of tosses of a fair coin until a head comes. – X={1, 2, 3, 4, 5, ……. . } ■ Number of people visiting an ATM machine in a day. – Y = {1, 2, 3, ……. } 9

Example of Continuous Random Variables ■ Interest centers around the proportion of people who respond to a certain mail order solicitation. Let X be that proportion. X is a random variable that takes on all values x for which 0 < x < 1. ■ Let X be the random variable defined by the: waiting time, in hours, between successive speeders spotted by a radar unit. The random variable X takes on all values x for which t > 0. 10

Discrete Probability Distribution ■ The set of ordered pairs (x, f(x)) is a probability function , probability mass function or probability distribution of discrete random variable x, if for each possible outcome x – f(x) ≥ 0 – f(x) = 1 – P(X = x) = f(x) 11

Cumulative Probability Distribution ■ Continuous probability distribution cannot be written in tabular form but it can be stated as a formula. Such a formula would necessarily be a function of the numerical values of the continuous random variable X and as such will be represented by the functional notation f(x). The function f(x) usually called probability density function or density function of X. 12

Continuous PDF ■ 13

Example ■ If a car agency sells 50% of its inventory of a certain foreign car equipped with airbags. Find a formula for the probability distribution of the number of cars with airbags among the next 4 cars sold by the agency. 14

Example ■ f( 0 )= 1/16, f( 1 ) = 1/4, f(2)= 3/8, f(3)= 1/4, and f( 4 )= 1/16. 15

Cumulative Distribution Function 16

Cumulative Distribution Function ■ 17

Example ■ 18

Cumulative Distribution Function ■ 19

Example of CDF ■ 20

Example of CDF ■ 21

Example of CDF ■ 22

Continuous Probability Distribution ■ Continuous probability distribution cannot be written in tabular form but it can be stated as a formula. Such a formula would necessarily be a function of the numerical values of the continuous random variable X and as such will be represented by the functional notation f(x). The function f(x) usually called probability density function or density function of X. 23

Continuous Probability Distribution ■ 24

Continuous Probability Distribution ■ 25

Exercise ■ Classify the following random variables as discrete or continuous: – X: the number of automobile accidents per year in Virginia. – Y: the length of time to play 18 holes of golf. – M: the amount of milk produced yearly by a particular cow. – N: the number of eggs laid each month by a hen. – P: the number of building permits issued each month in a certain city. – Q: the weight of grain produced per acre. 26

Exercise Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value w of W. 27

Exercise A coin is flipped until 3 heads in succession occur. List only those elements of the sample space that require 6 or less tosses. Is this a discrete sample space? 28

Exercise ■ Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X: ■ (a) f(x) = c(x 2 + 4), for x= 0, 1, 2, 3; 29

Exercise ■ Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X: 30

Exercise ■ The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function Find the probability that a bottle of this medicine will have a shelf life of (a) at least 200 days; (b) anywhere from 80 to 120 days. 31

Exercise ■ The proportion of people who respond to a certain mail-order solicitation is a continuous random variable X that has the density function (a) Show that P(0 < X < 1) = 1. (b) Find the probability that more than 1/4 but fewer than 1/2 of the people contacted will respond to this type of solicitation. 32

Exercise ■ The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given by ■ Construct the cumulative distribution function of X. 33

Exercise ■ The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given by ■ Construct the cumulative distribution function of X. 34

Exercise The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function Find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulative distribution function of X; (b) using the probability density function of X. 35

Joint Probability Distribution ■ 36

Joint Probability Distribution ■ 37

Continuous Joint PDF ■ 38

Continuous Joint PDF ■ 39

Marginal Distribution ■ 40

Marginal Distribution ■ Show that rows and columns of the previous problem are marginal distributions. 41

Marginal Distribution ■ The fact that the marginal distributions g(x) and h(y) are indeed the probability distributions of the individual variables X and Y alone can be verified by showing that the conditions of definitions of probability function are satisfied. ■ The set of ordered pairs (x, f(x)) is a probability function , probability mass function or probability distribution of discrete random variable x, if for each possible outcome x – f(x) ≥ 0 – f(x) = 1 – P(X = x) = f(x) 42

Conditional Distribution ■ 43

Conditional Distribution ■ 44

Conditional Distribution ■ 45

Conditional Distribution ■ 46