The Geometric Distribution: Some Notes

The Geometric Distribution| AP Statistics | SHUBLEKA    

Geometric Setting 1. Each observation falls into one of just two categories, which for convenience; we call “success” and “failure”. 2. The observations are all independent. 3. The probability of success, call it p, is the same for each observation 4. The variable of interest X is the number of trials required to obtain the first success. For data produced in a geometric setting, X is called a geometric random variable and its probability distribution is called geometric distribution. Examples: I. Flip a coin until you get a head. II. Roll a die until you get a 6. III. Attempt three-point shots until you make a basket. Calculating Probabilities

P ( X = n) = (1 − p ) n −1 p gives the probability of obtaining the first success in the n-th trial. X 1 P(X) p

2

(1 − p ) p

3

4

(1 − p) p 2

5

(1 − p ) p 3

6

(1 − p ) p 4

7

(1 − p ) p 5

8

(1 − p ) p 6

(1 − p )7 p

∞

Question: Is it true that

∑ P = 1? i =1

i

Example: Roll a die until a 6 occurs. Verify geometric setting with p = 1/6. TI-83: Compute 1/6. Multiply answer by 5/6. Hit enter a number of times to see additional probabilities. TI-83: geometcdf(p,n) gives P( X ≤ n) Expected Value and Other Properties of a Geometric Random Variable

μX = σ2 =

1 p

1− p p2

P ( X > n) = 1 − P( X ≤ n) = (1 − p) n Example: probability that it takes more than 6 rolls to get a six is

( 56 )

Next time: simulations of the geometric distribution and exercises.

 

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