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BOBBY B. LYLE SCHOOL OF ENGINEERING EMIS - SYSTEMS ENGINEERING PROGRAM SMU Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Discrete Probability Distributions Discrete Random Variables & Probability Distributions Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering 1 Random Variable 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 x1, x2, …, xk 2 Probability Mass Function Associated with a discrete random variable X having possible values x1, x2, …, xn 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. 3 Probability Mass Function The function p (x )is the probability mass function of the discrete random variable X if, for each possible outcome , x 1. p ( x) 0 2. p( x) 1 X 3. P( X x) p ( x) 4 Probability Distribution Function The (cumulative) probability distribution function, F (x), of a discrete random variable Xwith probability mass function p (x )is given by F ( x) P( X x) p(t ) t X 5 p(x) Probability Mass Function x 0 1 2 3 4 F(x) 1 Probability Distribution Function 0.5 0 x 0 1 2 3 4 6 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 7 Example Solution - Probability Mass Function and Probability Distribution Function 8 Mean or Expected Value of a Discrete Random Variable X • Mean or Expected Value of X μ EX xp(x) all x •Note: The interpretation of μ: The average of X in the long term. 9 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 10 Example Solution - Calculation of Mean 11 Variance & Standard Deviation of a Discrete Random Variable X • Variance – Definition Var X σ (x μ) p(x) 2 – Rule 2 all x Var X E X μ 2 2 x px μ 2 2 x • Standard Deviation σ Var(X) 12 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 13 Example – Family Planning In planning a family of 4 children, find the probability distribution of: a. b. X = the number of boys 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. 14 Discrete Uniform Distribution Definition - If the random variable X assumes the values x1, x2, ... xk with equal probabilities, then X has a discrete uniform distribution with probability mass function 1 p( x; k ) k for x x1 , x 2 , ... x k 15 Discrete Uniform Distribution If X has the discrete uniform distribution U(k), then the mean and variance are k Ex xi i 1 k k and 2 x i 1 2 i k 16 Rules 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 EaX b aμ b and Var aX b a Var X 2 17 Rules If Y = g(X) is a function of a discrete random variable X, then μ Y Eg x gx px all X 18 Chebyshev’s Theorem The probability that any random variable X will assume a value within k standard deviations of the mean is at least 1 1 2 , i.e., k P k X k 1 1 k 2 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. 19