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Date Created: 09/29/16
STAT 206 Chapter 5: Discrete Probability Distributions 5.1 Probability Distribution for a Discrete Variable RECALL: discrete variables have numerical values that arise from a counting process ● Probability distribution for a discrete variable: mutually exclusive list of all the possible numerical outcomes along with the probability of occurrence of each outcome ● Expected value of a random variable is the mean (μ) of its probability distribution ○ Expec ted value > E(x) = ΣxᵢP(xᵢ) ■ Σ = the sum ■ Xᵢ = a possible outcome ■ P(xᵢ) = probability of that outcome ■ The sub ‘i’ represents whatever value you’re working with ○ To calculate E(x), multiply each outcome by its corresponding probability, then add up all those values up (find the sum of the values) ○ EXAMPLE Customers per hour P(x) xᵢP(xᵢ) 0 0.10 (0)(0.10) = 0.00 1 0.25 (1)(0.25) = 0.25 2 0.50 (2)(0.50) = 1.00 3 0.15 (3)(0.15) = 0.45 Sum: 1.00 Sum = μ = E(x) = 1.70 ● Variance of a discrete variable = σ² ○ σ² = Σ[xᵢ﹣E(x)]²P(xᵢ) ○ To calculate σ², square the difference of each outcome and expected value, multiply that by the probability of said outcome, then add up all values ○ EXAMPLE Customers/hour P(x) xᵢP(xᵢ) [xᵢ﹣E(x)]²P(xᵢ) 0 0.10 (0)(0.10) = 0.00 (01.70)²(0.10) = 0.289 1 0.25 (1)(0.25) = 0.25 (11.70)²(0.25) = 0.123 2 0.50 (2)(0.50) = 1.00 (21.70)²(0.50) = 0.045 3 0.15 (3)(0.15) = 0.45 (31.70)²(0.15) = 0.254 Sum = μ = E(x) = 1.70 Sum = σ² = 0.711 ● Standard Deviation of a discrete variable = σ = √σ² (square root of the variance) ○ EXAMPLE σ² = 0.711 → σ = √σ² = √0.711 = 0.843 5.3 Binomial Distribution ● Mathematical model: a mathematical expression that represents a variable of interest; when one exists, you can compute the exact probability of occurrence of any particular value of the variable ○ Discrete random variables use a p robability distribution function ○ Binomial distribution: a mathematical model used in many business situations and when the discrete variable is the number of events of interest in a sample of n observations ○ Properties of the Binomial Distribution ■ Consists of a fixed number of observations: trials ● Ex: 15 tosses of a coin flip ■ Each observation is classified into one of two mutually exclusive and collectively exhaustive categories; categorized as to whether or not the “event of interest” occurred (considered a success) ■ The probability of an observation being classified as the event of interest, ????, is constant from observation to observation. Thus, the probability of an observation being classified as not being the event of interest, 1−????, is constant over all observations. ■ The value of any observation is independent of the value of any other observation. ● Counting technique for Binomial: Rule of Combinations ○ r = x (the outcome) ○ *0! = 1 by definition* ● Binomial Distribution: WHERE: ○ P(X = x | n, π) = probability that X = x events of interest, g and ○ n = number of observations ○ π = probability of an event of interest; also represented by ‘p’ ○ 1π = probability of not having an event of interest (sometimes 1π OR 1p = q) ○ x = number of events of interest in the sample ○ = number of combinations of x events of interest out of n observations ○ Excel function: BINOM.DIST(<#successes>,<#trials>,<probability_success>,<cumulative?>) ● Shape of the binomial distribution is controlled by the values o (p) and n ○ When π = 0.5, the binomial distribution is symmetrical ○ When π does NOT equal 0.5, the binomial distribution is skewed ○ The closer π is to 0.5 and the larger number of observation , the less skewed ● Binomial Distribution Characteristics ○ Mean: μ = E(x) = n???? ○ Variance: σ² = n????(1????) ○ Standard deviation: σ = √(n????(1????)) ○ Where:n = sample size (number of trials) ???? = probability of the event of interest (success) for any trial (1????) = probability of no success for any trial 5.4 Poisson Distribution ● Used when interested in the umber of times an event occurs in a given “area of opportunity” ○ Area of opportunity: interval of time (continuous unit of time), volume or area in which more than one occurrence of an event can occur ● Example situations: number of scratches in a car’s paint number of mosquito bites on a person number of people arriving at a bank ● Poisson Distribution: o calculate probabilities in situations like the ones above, IF: ○ You wish to count the number of times an event occurs in a given area of opportunity ○ Probability that an event occurs in one area of opportunity i s the same for all areas of opportunity ○ Number of events that occur in one area of opportunity is independent of the number of event that occur in the other areas of opportunity ○ Probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller ○ Average number of events per unit is λ (lambda) ● Poisson Distribution formula and characteristics ○ P(X=x | λ) = probability that X=x events in an area of opportunity given λ ○ λ = expected number of events ○ e = mathematical constant approximated by 2.71828 ○ x = number of events ● Mean = μ = λ ● Variance = σ² = λ ● Standard deviation = σ = √σ² = √λ ● Where λ = expected number of events ● Excel function: POISSON.DIST(number_successes,lambda,cumulative?) ○ Excel function: HYPEGEOM.DIST(<#successes>,<n>,<pop_#successes>,<N>,cumulative?)
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