GEOMETRIC DISTRIBUTION

ABSTRACT

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 ;

  • The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set .

Which of these is called 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 is the first success is

Pr(X=k)=(1-p)k-1p

for k = 1, 2, 3, 4, ….

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(Y=k)=Pr(X=k+1)=(1-p)kp

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.

INTRODUCTION

Consider a sequence of trials, where each trial has only two possible outcomes (designated failure and success). The probability of success is assumed to be the same for each trial. In such a sequence of trials, the geometric distribution is useful to model the number of failures before the first success since the experiment can have an indefinite number of trials until success, unlike the binomial distribution which has a set number of trials. The distribution gives the probability that there are zero failures before the first success, one failure before the first success, two failures before the first success, and so on.

When is the geometric distribution an appropriate model?

The geometric distribution is an appropriate model if the following assumptions are true.

  • The phenomenon being modeled is a sequence of independent trials.

  • There are only two possible outcomes for each trial, often designated success or failure.

  • The probability of success, p, is the same for every trial.

If these conditions are true, then the geometric random variable Y is the count of the number of failures before the first success. The possible number of failures before the first success is 0, 1, 2, 3, and so on. In the graphs above, this formulation is shown on the right.

An alternative formulation is that the geometric random variable X is the total number of trials up to and including the first success, and the number of failures is X − 1. In the graphs above, this formulation is shown on the left.

FORMULAS USED IN GEOMETRIC DISTRIBUTION

Usually, when you are given a geometric distribution you will be also given some formulas to find certain values of interest. Probability Mass Function

Since in a geometric distribution you are counting how many trials you take until getting a success, a natural question that arises is: What is the probability of getting the success in exactly x trials? This can be found by noting that, if you underwent x trials until you got the success, then you had x−1 failures, so

P(X=x) = (1-p)x-1p

where p is the probability of success, and 1−p is the probability of failure. You might also find this formula written as

P(X=x) = qx-1p,

where q=1−p.

Figure 1. Graph of the probability mass function of the geometric distribution

Cumulative Distribution Function

You can find a more realistic approach to an experiment by looking at the cumulative distribution function of the geometric distribution, which tells you the probability of getting success in x trials or less. For the geometric distribution, this is given by

P(X≤k)=1−(1−p)k.

Think of the stuffed bear example. Suppose you go to the claw machine with five spare quarters, the cumulative distribution function will tell you the probability of having at least one success with those five quarters, that is

P(X≤5)=1−(1−p)5.


Figure 2. Graph of the cumulative distribution function of the geometric distribution

APPLICATION

How is geometric progression applied in real life?

GP occurs in real life when each actor in a system behaves independently and is fixed. Examples include: If each person decides not to have another child depending on the current population, then annual population increase is geometric.

Each radioactive component disintegrates independently, resulting in a constant decay rate for each.

Interest rates, email chains, and so on are other instances. Geometric series are valuable because they may be used as a model for real-life circumstances.

Geometric sequences have a variety of applications in daily life, but one of the most prevalent is calculating interest. A term in a series is calculated by multiplying the first value in the sequence by a rate increased to the power of just less than the term number.

Applications of geometric Progression in real life

I’ll give you a handful of examples: When each person decides not to have another child based on the current population, population growth is geometric.

Use sequences in real life

Sequences are useful in both everyday life and higher mathematics. For example, sequences include the interest component of monthly payments made to pay off an automotive or home loan, as well as a month’s worth of maximum daily temperatures in one place. ## PROBLEMS

PROBLEMS

1: If the first term of a G.P. is 20 and the common ratio is 4. Find the 5th term.

Solution: Given,

First term, a=20

Common ratio, r=4

We know,

Nth term of G.P.,

an = arn-1

⇒ a5 = 20×44

= 20×256

= 5120

2.A patient suffers kidney failure and requires a transplant from a suitable donor. The probability that a random donor will match this patient’s requirements is 0.2.Suppose that no donor matches the patient’s requirements until a fifth donor comes in. What is the probability of this scenario?

Whenever you need to find the probability that the experiment requires an exact number of trials to succeed, you should start by writing its probability mass function. In this case, since p=0.2 then

Now, you can evaluate the above function when x=5, giving you

which means that the probability that this scenario happens is 8.192%.

CONCLUSION

  • The geometric distribution, also known as the geometric probability model, is a discrete probability distribution where the random variable X counts the number of trials performed until a success is obtained.

    • Since the least amount of trials required to obtain a success is 1, then the random variable X can take the values X=1,2,3,….

  • In order to model a situation using a geometric distribution, you need to make some assumptions: 1. There are only two possible outcomes of a trial, a success or a failure. 2. The trials are independent of each other. 3. The success probability remains unchanged trial after trial.

  • The formulas used in geometric distributions are the following:

    • The probability mass function is given byP(X=x)=(1−p)x−1p.

    • The cumulative distribution function isP(X≤k)=1−(1−p)k.

    • The expected value can be found asμ=1p.

    • The standard deviation isσ=1−pp2.

  • The exponential distribution is similar to the geometric distribution in the sense that both describe situations in which you are looking for the first success of a trial. However, the exponential distribution is a continuous distribution, while the geometric distribution is a discrete distribution.

REFERENCES

A modern introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005. pp. 48–50, 61–62, 152. ISBN 9781852338961. OCLC 262680588.:

//www.studysmarter.us/explanations/math/statistics/geometric-distribution/

https://unacademy.com/content/cat/study-material/mathematics/geometric-progression-is-used/