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Transcript of 1 Set #3: Discrete Probability Functions Define: Random Variable – numerical measure of the...
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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”
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Discrete Probability Functions
Probability Distribution may take form asTableGraphHistogramMathematical Formula
Probability Functions will also haveMeanExpected ValueVarianceStandard Deviation
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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 = 1
2. 0 ≤ P(x) ≤ 1 i.e. P(x) is between 0 and 1 inclusive
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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
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Distributions of Random Variables
Define: Mean of discrete random variable μx= sum of (value of random variable times the probability of observing the random variable x) μx = Σ [x*P(x)]
OR: mean of a discrete random variable is the average outcome if experiment is repeated many, many times
that is: – μx approaches zero
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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) = μx = Σ [x*P(x)]
OR: the expected value of a discrete random variable is the mean of that discrete random variable
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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
σ2x = Σ [(x - μx)2 *P(x)]
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Distributions of Random Variables
Define: Standard Deviation of a discrete random variable is the square root of the variance
σx = √σ2x
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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
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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
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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
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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
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Distributions of Random Variables
Computing the Probability of Binomial Experiments
P(x) = nCx px (1-p)n-x
For x = 0, 1, 2, . . . N
WoW this is a very powerful result
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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
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Distributions ofRandom Variables
1) Computing the Probability of Binomial ExperimentsP(x) = nCx px (1-p)n-x
For x = 0, 1, 2, . . . N
2) Mean (or Expected Value) = μx= n*p
3) Std deviation σx = √ n*p*(1-p)
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More Distributions of Random Variables
Geometric Probability Distribution
Hypergeometric Probability Distribution
Negative Binomial Probability Distribution
Poisson Probability Distribution
Note: These distributions will haveMeanStandard Deviation
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Distributions of Random Variables
Geometric Probability DistributionNumber of trials until success
Same requirements as Binomial Distr.
See page 357, problem # 58
Negative Binomial Probability DistributionTo compute the number of trials necessary
to observe “r” successes
Same requirements as Binomial Distr.
See page 357, problem # 59
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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
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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 µx = λt
Standard deviation = sq root (mean)e = base of the natural log system ~ 2.718281828
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Distributions of Random Variables
Poisson Probability Distribution Example #6.3.13 Insect Fragments legally in Peanut Butter
1. Calculate the Expected Value = mean µx = λt = rate * interval = 0.3 fragments * 5 gram sample = 1.5 fragments per sample = λt
2. Probability of exactly 2 fragments3. Probability of fewer than 2 frags4. Probability of at least 2 fragments
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Notes of the Day
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