Section 8.2: The Geometric Distribution

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Presentation transcript:

Section 8.2: The Geometric Distribution AP Statistics Section 8.2: The Geometric Distribution

Objective: To be able to understand and calculate geometric probabilities. Ex. Russian Roulette Criteria for a Geometric Random Variable: Each observation can be classified as a success or failure. p is the probability of success and p is fixed. The observations are independent. The random variable measures the number of trials necessary to obtain the first success. (it includes the 1st success)

Geometrical Probability Formula: If data are produced in a geometric setting and X = the number of trials until the first success occurs, then X is called a geometric random variable. Geometrical Probability Formula: Let X be a geometric random variable and n be the number of trials until we obtain our first success. If X~G(p), then 𝑃 𝑋=𝑛 = 𝑞 (𝑛−1) ∙𝑝 The geometric probability distribution is as follows: 𝑿= 𝒙 𝒊 1 2 3 … n P(𝑿= 𝒙 𝒊 ) 𝒑 𝒒𝒑 𝒒 𝟐 𝒑 𝒒 𝒏−𝟏 ∙𝒑 F(𝑿= 𝒙 𝒊 ) 𝒑+𝒒𝒑 𝒑+𝒒𝒑+ 𝒒 𝟐 𝒑 𝑝 𝑛

Points: This is an infinite distribution. The smallest X can be is 1. Every geometric distribution is __________________. The cumulative distribution is always ___________________. Calculator notation: For P(X = k) --- use geometpdf(p,X) For P(X ≤ k) --- use geometcdf(p,X) How can we show that the sum of the terms of an infinite distribution sum to 1?

Ex. Suppose you work at a blood bank and are interested in collecting type A blood. It is known that 15% of the population is type A. Let X represent the number of donors until and including the first type A donor is found. Does this example meet the criteria for a geometric setting? Find probability that the first type A donor is the 4th donor of the day. Find probability that the first type A donor is the 2nd donor of the day. Find probability that the first type A donor occurs before the 4th donor of the day.

Find probability that the first type A donor is at least the 5th donor of the day. Complete the probability distribution for the geometric random variable X for the first 6 terms. 𝑿= 𝒙 𝒊 P(𝑿= 𝒙 𝒊 ) F(𝑿= 𝒙 𝒊 )

If X ~ G(p), then 𝜇 𝑋 = 1 𝑝 and 𝜎 2 𝑋 = (1−𝑝) 𝑝 2 If X ~ G(p), then 𝜇 𝑋 = 1 𝑝 and 𝜎 2 𝑋 = (1−𝑝) 𝑝 2 . Two Methods for Solving P(X > n): 1. 2. Proof of method #2: