Geometric PD Geometric Distribution If you were to

Geometric PD

Geometric Distribution If you were to flip a coin wanting to get a head, you would keep flipping until you obtained that head.

Geometric Distribution Newborns Revisited. . . Suppose we were not interested in the number of females born out of the next five births, but which birth would result in the first female being born? How is this question different from a binomial distribution?

Definition of the GPD • The way it works is, keep conducting trials until the first success comes up. • GPD measures the probability of the first success.

1. An experiment consists of repeating trials until first success. 2. Each trial has two possible outcomes; (a)A success with probability p (b) A failure with probability q = 1 − p. Admitted or rejected on your college app 3. Repeated trials are independent. X = number of trials to first success

Properties of Geometric Distributions: • There are two mutually exclusive outcomes that result in a success or failure • Each trial is independent of the others • The probability of success is the same for all trials. * We repeat untill we get sucess A geometric random variable x is defined as x = the number of trials until the FIRST success is observed ( including the success).

x 1 2 3 4 . . . So what are the possible values of x To infinity How far will this go?

Formula for GPD ?

Example-1: If you were to flip a die wanting to get a #3, you would keep flipping until you obtained that #3. What is the probability of getting the first head on the first attempt? 2 nd attempt? 5 th attempt?

• If the trials are repeated k times until the first success, we will have had k – 1 failures.

Probability Formula for the Geometric Distribution Let p = constant probability that any trial results in a success Where r = 1, 2, 3, …

* When we studied the Binomial distribution, we were only interested in the probability for a success or a failure to happen. ** The geometric distribution addresses the number of trials necessary before the first success. *** is Probability of first success **** If the trials are repeated k times until the first success, we will have had k – 1 failures.

Examples of Geometric PDF • Passing the SAT test at your 3 rd time taking it • First plane arriving at an airport that needs repair • Being accepted at the sixth time you tell someone you like her • First SFLS’s student getting offer from MIT

Example-2: Suppose that 40% of students who drive to campus at your school or university carry jumper cables(跨接��). Your car has a dead battery and you don’t have jumper cables, so you decide to stop students as they are headed to the parking lot and ask them whether they have a pair of jumper cables. Let x = the number of students stopped before finding one with a pair of jumper cables Is this a geometric distribution? Yes

Let x = the number of students stopped before finding one with a pair of jumper cables p = 0. 4 What is the probability that third student stopped will be the first student to have jumper cables? P(x = 3) = (. 6)2(. 4) =. 144 What is the probability that at most three student are stopped before finding one with jumper cables? P(x < 3) = P(1) + P(2) + P(3) = (. 6)0(. 4) + (. 6)1(. 4) + (. 6)2(. 4) =. 784

How to use GDC?

1. 2. 3. 4. TI-nspire Menu Probability (5) Distributions (5) Geometric pdf (F)………(to find prob of exactly one value) Or 5. Geometric cdf (G)----(to find multiple Geomteric prob like at most, at least problems )

Try Me Over a very long period of time, it has been noted that on Friday’s 30% of the customers at the drive-in window at the bank make deposits. What is the probability that the 5 th customer is the first one made the deposit?

Mean of GPD Mean: E(x)= =

• Proof:

Mean and Standard Deviation of GPD Mean: = Variance: 2 = Standard Deviation: =

Last Example: If a production line has a 20% defective rate. What is the average number of inspections to obtain the first defective?

Quick survey I feel I understand “GPD” a) Very well b) With some review, I’ll be good c) Not really d) Not at all