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Chapter 5: Discrete Probability Distribution Section 1: Probability Distributions Random variable: variable whose values are determined by chance. Discrete variable: variable that can assume only a specific number of values and have a finite number of possible values or a infinite number of values that can be counted. Discrete probability distribution: consists of the values a random variable can assume and the corresponding probabilities of the values. *Discrete probability can be shown by using a graph or a table. Section 2: Mean, Variance, Standard Deviation, and Expectation Formula for the Mean of a Probability Distribution: The mean of a random variable with a discrete probability distribution is
𝝁 = 𝑿𝟏 ∙ 𝑷 𝑿𝟏 + 𝑿𝟐 ∙ 𝑷 𝑿𝟐 + 𝑿𝟑 ∙ 𝑷 𝑿𝟑 +⋯+ 𝑿𝒏 ∙ 𝑷 𝑿𝒏 = 𝚺(𝑿 ∙ 𝑷 𝑿 )
𝑋!,𝑋!,𝑋!,… ,𝑋! are the outcomes 𝑃(𝑋!), 𝑃 𝑋! , 𝑃 𝑋! ,… , 𝑃(𝑋!) are the corresponding probabilities.
Note: Σ(𝑋 ∙ 𝑃 𝑋 ) means to sum the products. (Get the products and then add). Formula for the Variance of a Probability Distribution: The formula for the variance of a probability distribution is:
𝝈𝟐 = [𝑿𝟏𝟐 ∙ 𝑷 𝑿𝟏 + 𝑿𝟐𝟐 ∙ 𝑷 𝑿𝟐 + 𝑿𝟑𝟐 ∙ 𝑷 𝑿𝟑 +⋯+ 𝑿𝒏𝟐 ∙ 𝑷 𝑿𝒏 ] − 𝝁𝟐
= 𝚺 𝑿𝟐 ∙ 𝑷 𝑿 − 𝝁𝟐
Note: Σ[𝑋! ∙ 𝑃 𝑋 ] means to sum the products. (Get the products then add).
The standard deviation of a probability distribution is
𝝈 = 𝝈𝟐 = 𝚺 𝑿𝟐 ∙ 𝑷 𝑿 − 𝝁𝟐 Expected Value of a discrete random variable of a probability distribution is the theoretical average of the variable.
𝝁 = 𝑬 𝑿 = 𝚺𝑿 ∙ 𝑷 𝑿
𝑬 𝑿 is the expected value.
Rounding rule for the mean, variance and standard deviation for a probability distribution Mean, variance, and standard deviation should be rounded off to one more decimal than the outcome X. When fractions are used, they should be reduced to lowest terms. ***When expected value problems involve money, round off to the nearest cent.
Chapter 5: Probability Distribution Section 3: The Binomial Distribution Binomial experiment: a probability experiment that satisfies the following four requirements:
1. There must be a fixed number of trials. 2. Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These
outcomes can be considered as either success or failure. 3. The outcomes of each trial must be independent of one another. 4. The probability of a success must remain the same for each trial.
Binomial distribution: the outcomes of a binomial experiment and the corresponding probabilities of these outcomes. Notation for the Binomial Distribution 𝑷 𝑺 Probability of Success
𝑷 𝑭 Probability of Failure
𝒑 Numerical probability of a success
𝒒 Numerical probability of a failure
𝑃 𝑆 = 𝑝 𝑎𝑛𝑑 𝑃 𝐹 = 𝑞
𝒏 Number of trials
𝑿 Number of successes in n trials
0 ≤ 𝑋 ≤ 𝑛 𝑎𝑛𝑑 𝑋 = 0, 1, 2, 3,… ,𝑛
Binomial Probability Distribution In a binomial experiment, the probability of exactly X successes in n trials is:
𝑷 𝑿 = 𝒏!
𝒏 − 𝑿 !𝑿!∙ 𝒑𝑿 ∙ 𝒒𝒏!𝑿
Mean for the Binomial Distribution 𝝁 = 𝒏 ∙ 𝒑
Variance for the Binomial Distribution
𝝈𝟐 = 𝒏 ∙ 𝒑 ∙ 𝒒 Standard Deviation for the Binomial Distribution
𝝈 = 𝝈𝟐 = 𝒏 ∙ 𝒑 ∙ 𝒒
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