1 Set #3: Discrete Probability Functions Define: Random Variable – numerical measure of the outcome of a probability experiment Value determined by chance Denoted using letters, such as X Recall and extend: differences between “discrete random variable” and “continuous random variable”

2 Discrete Probability Functions Probability Distribution may take form as Table Graph Histogram Mathematical Formula Probability Functions will also have Mean Expected Value Variance Standard Deviation

3 Rules for Discrete Probability Function Let P(x) denote the probability that a random variable X equals x; then 1. Σ P(x) = 1 i.e.: Sum of all probabilities of x = ≤ P(x) ≤ 1 i.e. P(x) is between 0 and 1 inclusive

4 Distributions of Random Variables Define: Probability Histogram – a histogram where x-axis corresponds to value of random variable y-axis represents the probability of each value of the discrete random variable Construct Probability Histogram the same way as a relative frequency histogram, except vertical axis is probability

5 Distributions of Random Variables Define: Mean of discrete random variable μ x = sum of (value of random variable times the probability of observing the random variable x) μ x = Σ [x*P(x)] OR: mean of a discrete random variable is the average outcome if experiment is repeated many, many times that is:  – μ x approaches zero

6 Distributions of Random Variables Define: Expected Value E(x) is the sum of (value of random variable times probability of observing that random variable x) E(X) = μ x = Σ [x*P(x)] OR: the expected value of a discrete random variable is the mean of that discrete random variable

7 Distributions of Random Variables Define: Variance of a discrete random variable - is the weighted average of the squared deviations where the weights are the probabilities σ 2 x = Σ [(x - μ x ) 2 *P(x)]

8 Distributions of Random Variables Define: Standard Deviation of a discrete random variable is the square root of the variance σ x = √σ 2 x

9 Distributions of Random Variables Define: trial - each repetition of an experiment. Define: disjoint outcome - two mutually exclusive outcomes Typically the disjoint outcomes are called “success” and “failure” Probability of success = One minus probability of failure

10 Distributions of Random Variables Distributions of Discrete Random Variables Thus far have had: Mean Expected Value Variance Standard Deviation Will have next: Binomial Probability Function Poisson Probability Function And... Distributions of Continuous Random Variables

11 Distributions of Random Variables Binomial Probability Experiment Criteria 1. Experiment performed fixed number of times (i.e. trials discrete) 2. Trials are independent 3. For each trial there are two disjoint outcomes: success & failure 4. The probability of success is the same for each trial of the experiment

12 Distributions of Random Variables Binomial Probability Experiment Notations: n is the number of independent trials p is the probability of success x is number of successes in n independent trials

13 Distributions of Random Variables Computing the Probability of Binomial Experiments P(x) = n C x p x (1-p) n-x For x = 0, 1, 2,... N WoW WoW this is a very powerful result

14 Distributions of Random Variables Keep on going along this trail Construct Binomial Probability Histogram Use mean, standard deviation, and empirical rule to check for unusual results

15 Distributions of Random Variables 1) Computing the Probability of Binomial Experiments P(x) = n C x p x (1-p) n-x For x = 0, 1, 2,... N 2) Mean (or Expected Value) = μ x = n * p 3) Std deviation σ x = √ n * p * (1-p)

16 More Distributions of Random Variables Geometric Probability Distribution Hypergeometric Probability Distribution Negative Binomial Probability Distribution Poisson Probability Distribution Note: These distributions will have Mean Standard Deviation

17 Distributions of Random Variables Geometric Probability Distribution Number of trials until success Same requirements as Binomial Distr. See page 357, problem # 58 Negative Binomial Probability Distribution To compute the number of trials necessary to observe “r” successes Same requirements as Binomial Distr. See page 357, problem # 59

18 Distributions of Random Variables Poisson Process Conditions: 1. Probability of two or more successes in any sufficiently small subinterval is zero 2. Probability of success is the same of any two intervals of equal length 3. Number of successes in any interval is independent of number of successes in any other interval

19 Distributions of Random Variables Poisson Probability Distribution Function: for X number of successes in an interval of fixed length t, and average number of occurrences λ (lamba) P(x) = [(λt) x /x!] e -λt Expected value equals mean µ x = λt Standard deviation = sq root (mean) e = base of the natural log system ~

20 Distributions of Random Variables Poisson Probability Distribution Example # Insect Fragments legally in Peanut Butter 1. Calculate the Expected Value = mean µ x = λt = rate * interval = 0.3 fragments * 5 gram sample = 1.5 fragments per sample = λt 2. Probability of exactly 2 fragments 3. Probability of fewer than 2 frags 4. Probability of at least 2 fragments

21 Notes of the Day