1. 1 Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 Discrete Probability Distributions Discrete Random Variables & Probability Distributions

2. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 2 Definition - A random variable is a mathematical function that associates a number with every possible outcome in the sample space S. Notation - Capital letters, usually X or Y, are used to denote random variables. Corresponding lower case letters, x or y, are used to denote particular values of the random variables X or Y. Definition - A discrete random variable X is a random variable that can take on or assume a finite number of possible values, say x 1, x 2, …, x k Random Variable

3. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 3 Associated with a discrete random variable X having possible values x 1, x 2, …, x n is a function called the probability mass function. The probability mass function of X associates with each possible value of X the probability of its occurrence. This set of ordered pairs, each of the form, (value of x, probability of that value occurring) or ( x, p(x) ) is the probability mass function of X. Probability Mass Function

4. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 4 The function is the probability mass function of the discrete random variable if, for each possible outcome, 1. 2. 3. Probability Mass Function

5. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 5 The (cumulative) probability distribution function,, of a discrete random variable with probability mass function is given by Probability Distribution Function

6. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 6 p(x) x F(x) x 1 0.5 0 Probability Mass Function Probability Distribution Function 0 1 2 3 4

7. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 7 Example - Probability Mass Function and Probability Distribution Function If an experiment is “Toss a coin 3 times in sequence” and the random variable X is defined to be the number of heads that result, determine and plot the probability mass function and probability distribution function for X if (a)The coin is fair (b)The coin is biased with P(H)=0.75

8. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 8 Example Solution - Probability Mass Function and Probability Distribution Function

9. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 9 Mean or Expected Value of X Note: The interpretation of μ: The average of X in the long term. Mean or Expected Value of a Discrete Random Variable X

10. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 10 Example-Calculation of Mean If an experiment is “Toss a coin 3 times in sequence” and the random variable X is defined to be the number of heads that result, what is the mean or expected value of X if (a)The coin is fair (b)The coin is biased with P(H)=0.75

11. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 11 Example Solution - Calculation of Mean

12. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 12 Variance – Definition – Rule Standard Deviation Variance & Standard Deviation of a Discrete Random Variable X

13. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 13 Example-Calculation of Standard Deviation If an experiment is “Toss a coin 3 times in sequence” and the random variable X is defined to be the number of heads that result, what is the standard deviation of X if (a)The coin is fair (b)The coin is biased with P(H)=0.75

14. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 14 In planning a family of 4 children, find the probability distribution of: a.X = the number of boys b.Y = the number of changes in sex sequence Find (i) the probability mass and distribution functions (and plot), (ii) the mean, (iii) the variance, and (iv) the standard deviation. Example – Family Planning

15. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 15 Definition - If the random variable X assumes the values x 1, x 2,... x k with equal probabilities, then X has a discrete uniform distribution with probability mass function Discrete Uniform Distribution

16. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 16 If X has the discrete uniform distribution U(k), then the mean and variance are and Discrete Uniform Distribution

17. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 17 If a and b are constants and if  = E(X) is the mean and  2 = Var(X) is the variance of the random variable X, respectively, then and Rules

18. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 18 If Y = g(X) is a function of a discrete random variable X, then Rules

19. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 19 The probability that any random variable X will assume a value within k standard deviations of the mean is at least, i.e., Remark: Chebyshev’s Theorem gives a conservative estimate of the probability that a random variable assumes a value within k standard deviations of its mean for any real number k. Chebyshev’s Theorem