# What is Geometric distribution: Definition and 29 Discussions

In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:

The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ... }
The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }Which of these one calls "the" geometric distribution is a matter of convention and convenience.
These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one (distribution of the number X); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.
The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is

Pr
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X
=
k
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=
(
1

p

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k

1

p

{\displaystyle \Pr(X=k)=(1-p)^{k-1}p}
for k = 1, 2, 3, ....
The above form of the geometric distribution is used for modeling the number of trials up to and including the first success. By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success:

Pr
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Y
=
k
)
=
Pr
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X
=
k
+
1
)
=
(
1

p

)

k

p

{\displaystyle \Pr(Y=k)=\Pr(X=k+1)=(1-p)^{k}p}
for k = 0, 1, 2, 3, ....
In either case, the sequence of probabilities is a geometric sequence.
For example, suppose an ordinary die is thrown repeatedly until the first time a "1" appears. The probability distribution of the number of times it is thrown is supported on the infinite set { 1, 2, 3, ... } and is a geometric distribution with p = 1/6.
The geometric distribution is denoted by Geo(p) where 0 < p ≤ 1.

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1. ### I Geometric Distribution Problem Clarification

(Geometric). The probability of being seriously injured in a car crash in an unspecified location is about .1% per hour. A driver is required to traverse this area for 1200 hours in the course of a year. What is the probability that the driver will be seriously injured during the course of the...
2. ### Geometric Distribution: Finding Specific p Value for Mean Calculation

I know the p.g.f. of X is $$q/(1-ps)$$ and that the mean is $$p/q$$, but how do I find a specific value for p here?
3. ### I How to understand this property of Geometric Distribution

There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$. I understand it in such a way: ##X## is independent, that's to say after there are ##(n+k-1)## successive failures, ##k## additional trials performed afterward won't be impacted, so these ##k##...
4. ### Moments from characteristic function geometric distribution

Homework Statement Hi, I have the probabilty density: ##p_{n}=(1-p)^{n}p , n=0,1,2... ## and I am asked to find the characteristic function: ##p(k)= <e^{ikn}> ## and then use this to determine the mean and variance of the distribution. Homework Equations [/B] I have the general expression...
5. ### Proof that an interval is a confidence interval for Geom(q)

Hello Physicsforum Homework Statement I have a problem proving this: Given C(x)=[0, 3/x] for all x\in\chi, with \chi=\Omega being the sample space and P_q=Geom(q) being the geometric distribution. I have to show that C(x) is a confidence Interval for q but I don't know how to get started...
6. ### Markov's Inequality for Geometric Distribution.

Homework Statement Let X∼Geometric(p). Using Markov's inequality find an upper bound for P(X≥a), for a positive integer a. Compare the upper bound with the real value of P(X≥a). Then, using Chebyshev's inequality, find an upper bound for P(|X - EX| ≥ b). Homework Equations P(X≥a) ≤ Ex / a...
7. ### S1 Probability - Binomial & Geometric Distribution

Homework Statement Four players play a board game which requires them to take it in turns to throw two fair dice. Each player throws the two dice once in each round. When a double is thrown the player moves forward six squares. Otherwise the player moves forward one square Homework Equations...
8. ### Geometric distribution Problem

Homework Statement a man draws balls from an infinitely large box containing either white and black balls , assume statistical independence. the man draws 1 ball each time and stops once he has at least 1 ball of each color . if the probability of drawing a white ball is p , and and q=1-p is...
9. ### How to solve a geometric distribution problem with a biased coin?

A boy is playing with a biased coin. The probabilty of obtaining a head with the coin is 0.4. Determine the probability that the boy will require at least eleven tosses before obtaining his third head. I have been trying but can't get it at all... Can someone please explain me how to solve...
10. ### Geometric Distribution Probability problem

Homework Statement We roll a fair die until we get a three or a four. Z denotes the number of rolls needed. What is the probability that Z >= 3? (replacement assumed) Homework Equations Geometric distribution seems logical here? The Attempt at a Solution Let p(A) = p(getting a...
11. ### Fitting a geometric distribution to data

Let's say I have a series of 100 coin tosses, heads or tails. In fact (for my actual data) I don't know if subsequent trials are correlated or what the actual probabilities of getting heads or tails are. Nevertheless, I want to fit a geometric distribution, which gives me the distribution of the...
12. ### Discrete Random Variables - Geometric Distribution

Hi Guys, Long time reader first time poster... This simple question has stumped me all day and I think I've finally cracked it! I'm hoping someone can confirm that, or tell me how wrong I am - either is fine :) One in 1000 cows have a rare genetic disease. The disease is not contagious...
13. ### Geometric Distribution

Question: Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines execute one run, then both execute a second run, and so on). On each run, M1 fails with probability p1 and M2 with probability p2, all failure events being independent. Let the random...
14. ### Geometric Distribution

Question: Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines execute one run, then both execute a second run, and so on). On each run, M1 fails with probability p1 and M2 with probability p2, all failure events being independent. Let the random...
15. ### Median geometric distribution

Homework Statement How do you find the median of the geometric distribution? Homework Equations M is median if P(X>=M) >= 1/2 and P(X<=M)>=1/2. The Attempt at a Solution I have found this inequality using the geometric series: (m-1)*log(1-p) >= 1/2
16. ### Computing the Mean of a Geometric Distribution

Homework Statement Problem H-10. We will compute the mean of the geometric distribution. (Note: It's also possible to compute E(X^2) and then Var(X) = E(X^2)−(E(X))^2 by steps similar to these.) (a) Show that E(X) = (k=1 to infinity summation symbol) (k *q^k−1* p) where q = 1−p. (b)...
17. ### Geometric Distribution Coin Flip

Consider the following experiment: a coin that lands heads with probability p is flipped once; if on this first flip it came up H, it is then repeatedly flipped until a T occurs; else, if on the first...
18. ### Statistics: geometric distribution proof problem

Statistics: geometric distribution "proof" problem Homework Statement If Y has a geometric distribution with success probability p, show that: P(Y = an odd integer) = \frac{p}{1-q^{2}} Homework Equations p(y)=p(q)^{2} The Attempt at a Solution p(1)=pq^0 p(3)=pq^2 p(5)=pq^4...
19. ### Geometric Distribution Question

Homework Statement An experiment consistion of tossing three fair coins is performed repeatedly and "success" is when all three show a head. What is the probability that the success is on the third performance of the experiment? Homework Equations Geometric distribution equation p(x) =...
20. ### Geometric Distribution problem

Question: If Y has a geometric distribution with success probability .3, what is the largest value, y0, such that P(Y > y0) ≥ .1? Attempt: So i represented the probability of the random variable as a summation Sum from y0= y0+1 to infinity q^(yo+1)-1 p ≥ .1 using a change of variables...
21. ### Geometric distribution problem

Can anyone solve this for me? I think it is geometric distribution. Tom, Dick and Harry play .the following game. They toss a fair coin in turns. First Tom tosses, then Harry, then Dick, then Tom again and so on until one of them gets a Head and so becomes the winner. What is the...
22. ### Sufficient Estimator for a Geometric Distribution

Homework Statement Let X1,..,Xn be a random sample of size n from a geometric distribution with pmf P(x; \theta) = (1-\theta)^x\theta. Show that Y = \prod X_i is a sufficient estimator of theta. Homework Equations The Attempt at a Solution So \prod P(x_i, \theta) =...
23. ### Estimating Geometric Distribution

Homework Statement We return to the example concerning the number of menstrual cycles up to pregnancy, where the number of cycles was modeled by a geometric random variable. The original data concerned 100 smoking and 486 nonsmoking women. For 7 smokers and 12 nonsmokers, the exact number of...
24. ### Understanding Geometric Distribution

Geometric Distribution? Geometric Distribution: In an experiment, a die is rolled repeatedly until all six faces have finally shown.? What is the probability that it only takes six rolls for this event to occur? ANSWER = 0.0007716 What is the expected waiting time for this event to occur...
25. ### Geometric Distribution and probability

Homework Statement Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting, that is, of not obtaining a busy signal. Assume that your calls are independent. What is the probability that it requires more than five calls for you to connect...
26. ### Geometric Distribution, Poisson

The problem is the following; N has a geometric distribution with Pr(N=0)>0. M has a Poisson distribution. You are given: E(N) = E(M); Var(N) = 2Var(M) Calculate Pr (M>1). From general knowledge we know that the expected value of a variable in a geometric distribution E(N) =...
27. ### Geometric distribution problem

a couple decides that they will have kids until a girl is born. the outcome of each birth is independent event, and the probability that a girl will be born is 1/2. The birht at which the first girl appears is a geometric distribution. what is the expected family size. ok, so we know that...
28. ### Married couples - geometric distribution

A couple plans to continue having children until they have their first girl. Suppose the probability that a child is a girl is 0.5, the outcome of each birth is an independent event, and the birth at which the first girl appears has a geometric distribution. What is the couple's expected...
29. ### Primes and the Geometric Distribution

Given the probability of flipping a heads with a fair coin is \frac{1}{2}, what is the probability that the first heads occurs on a prime number?