Discrete Probability Distribution

The Poisson Probability Distribution is discrete and deals with events that can only take on specific, whole number values (like the number of cars passing a certain point in an hour). Poisson Probability Distribution models the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence ($\mu$). The events must be independent of each other and occur randomly.

The Poisson probability function gives the probability for the number of events that occur in a given interval (often a period of time) assuming that events occur at a constant rate during the interval.

Poisson Random Variable

The Poisson random variable satisfies the following conditions:

• The number of successes in two disjoint time intervals is independent
• The probability of success during a small time interval is proportional to the entire length of the time interval.
• The probability of two or more events occurring in a very short interval is negligible.

Apart from disjoint time intervals, the Poisson random variable is also applied to disjoint regions of space.

Applications of Poisson Probability Distribution

The following are a few of the applications of Poisson Probability Distribution:

• The number of deaths by horse kicking in the Prussian Army (it was the first application).
• Birth defects and genetic mutations.
• Rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent), especially in legal cases.
• Car accidents
• Traffic flow and ideal gap distance
• Hairs found in McDonald’s hamburgers
• Spread of an endangered animal in Africa
• Failure of a machine in one month

The formula of Poisson Distribution

The probability distribution of a Poisson random variable $X$ representing the number of successes occurring in a given time interval or specified region of space is given by

\begin{align*}
P(X=x)&=\frac{e^{-\mu}\mu^x}{x!}\,\,\quad x=0,1,2,\cdots
\end{align*}

where $P(X=x)$ is the probability of $x$ events occurring, $e$ is the base of the natural logarithm (~2.71828), $\mu$ is the mean number of successes in the given time interval (or region of space), $x$ is the number of events we are interested in, and $x!$ is the factorial of $x$.

Mean and Variance of Poisson Distribution

If $\mu$ is the average number of successes occurring in a given time interval (or region) in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to $\mu$. That is,

\begin{align*}
E(X) &= \mu\\
V(X) &= \sigma^2 =\mu
\end{align*}

A Poisson distribution has only one parameter, $\mu$ is needed to determine the probability of an event. For binomial experiments involving rare events (small $p$) and large values of $n$, the distribution of $X=$ the number of success out of $n$ trials is binomial, but it is also well approximated by the Poisson distribution with mean $\mu=np$.

When to Use Poisson Probability Distribution

The Poisson distribution is useful in various scenarios:

• Modeling Rare Events: Like accidents, natural disasters, or equipment failures.
• Counting Events in a Fixed Interval: Such as the number of customers arriving at a store in an hour, or the number of calls to a call center in a minute.
• Approximating the Binomial Distribution: When the number of trials ($n$) is large and the probability of success ($p$) is small.

It is important to note that

• The Poisson distribution is related to the exponential distribution, which models the time between events.
• It is a fundamental tool in probability theory and statistics, with applications in fields like operations research, queuing theory, and reliability engineering.

R and Data Analysis, Test Preparation MCQs

Frequently Asked Questions about Poisson Distribution

1. What is the Poisson Random Variable?
2. What is Poisson Probability Distribution?
3. Write the Formula of Poisson Probability Distribution.
4. Poisson distribution is related to what distribution?
5. Give some important applications of Poisson Distribution.
6. Describe the general situations in which Poisson distribution can be used.
7. Name the distribution that has equal mean and variance.
8. What are the required conditions for poison random variables?

A probability distribution for a discrete random variable $X$ is a list of each possible value for $X$ with the probability that $X$ will have that value when the experiment is run. The likelihood for the probability distribution of a discrete random variable is denoted by $P(X=x)$. The probability distribution of a discrete random variable is also called a discrete probability distribution.

A discrete probability distribution is a mathematical function that assigns probabilities to each possible value of a discrete random variable.

Example of Probability Distribution of a Discrete Random Variable

Let $X$ be a random variable representing the number of trials obtained when a coin is flipped three times in an experiment. The sample space of the experiment is:

$$HHH, HHT, HTH, THH, HTT, TTH, THT, TTT$$

where $T$ represents the occurrence of Tail and $H$ represents the occurrence of Head in the above experiment.

Then $X$ has 4 possible values: $0, 1, 2, 3$ for the occurrence of head or tail. The probability distribution for $X$ is given as below:

$X$	$P(X)$
0	$\frac{1}{8}$
1	$\frac{3}{8}$
2	$\frac{3}{8}$
3	$\frac{1}{8}$
Total	$1.0$

In a statistics class of 25 students are given a 5-point quiz. 3 students scored 0; 1 student scored 1, 4 students scored 2, 8 students scored 3, 6 students scored 4, and 3 students scored 5. If a student is chosen at random, and the random variable $S$ is the student’s Quiz Score then the discrete probability distribution of $S$ is

$S$	$P(S)$
0	0.12
1	0.04
2	0.16
3	0.32
4	0.24
5	0.12
Total	1.0

Note that for any discrete random variable $X$, $0\le P(X) \le 1$ and $\Sigma P(X) =1$.

Finding Probabilities from a Discrete Probability Distribution

Since a random variable can only take one value at a time, the events of a variable assuming two different values are always mutually exclusive. The probability of the variable taking on any number of different values can thus be found by simply adding the appropriate probabilities.

Mean or Expected Value of a Discrete Random Variable

The mean or expected value of a random variable $X$ is the average value that one should expect for $X$ over many trials of the experiment in the long run. The general notation of the mean or expected value of a random variable $X$ is represented as

$$\mu_x\quad \text{ or } E[X]$$

The mean of a discrete random variable is computed using the formula

$$E[X]=\mu_x = \Sigma x\cdot P(X)$$

Example 1

From the above experiment of three Coins the Expected value of the random variable $X$ is

$X$	$P(X)$	$x.P(X)$
0	$\frac{1}{8}$	$0 \times \frac{1}{8} = 0$
1	$\frac{3}{8}$	$1 \times \frac{3}{8} = \frac{3}{8}$
2	$\frac{3}{8}$	$2 \times \frac{3}{8} = \frac{6}{8}$
3	$\frac{1}{8}$	$3 \times \frac{1}{8} = \frac{3}{8}$
Total	$1.0$	$\frac{3}{2} = 1.5$

Thus if three coins are flipped a large number of times, one should expect the average number of trials (per 3 flips) to be about 1.5.

Example 2

Similarly, the mean of the random variable $S$ from the above example is

$S$	$P(S)$	$S\cdot P(S)$
0	0.12	$0 \times 0.12 = 0$
1	0.04	$1 \times 0.04 = 0.04$
2	0.16	$2 \times 0.16 = 0.32$
3	0.32	$3 \times 0.32 = 0.96$
4	0.24	$4\times 0.24 = 0.96$
5	0.12	$5 \times 0.12 = 0.60$
Total	$1.0$	$2.88$

Note that $2.88$ is the class average on the statistics quiz as well.

Variance and Standard Deviation of a Random Variable

One may be interested to find how much the values of a random variable differ from trial to trial. To measure this, one can define the variance and standard deviation for a random variable $X$. The variance of $X$ random variable is denoted by $\sigma^2_x$ while the standard deviation of the random variable $X$ is just the square root of $\sigma^2_x$. The formulas of variance and standard deviation of a random variable $X$ are:

\begin{align*}
\sigma^2_x &= \Sigma (x – \mu)^2 P(X)\\
\sigma_x &= \sqrt{\Sigma (x – \mu)^2 P(X)}
\end{align*}

Note that the standard deviation estimates the average difference between a value of $x$ and the expected value.

Calculating the Variance and Standard Deviation

The calculation of standard deviation for a random variable is similar to the calculation of weighted standard deviation in a frequency table. The $P(x)$ can be thought of as the relative frequency of $x$. The computation of variance and standard deviation of a random variable $X$ can be made using the following steps:

1. Compute $\mu_X$ (mean of the random variable)
2. Subtract the mean/average from each of the possible values of $X$. These values are called the deviations of the $X$ values.
3. Square each of the deviations calculated in the previous step.
4. Multiply each squared deviation (calculated in step 3) by the corresponding probability $P(x)$.
5. Sum the results of step 4. The variance of the random variable will be obtained representing $\sigma^2_X$.
6. Take the square root of the $\sigma^2_X$ computed in Step 5.

Importance of Discrete Probability Distributions

• Modeling Real-World Phenomena: Discrete Distributions help us understand and model random events in various fields of life such as engineering, finance, and the sciences.
• Decision Making: These distributions provide a framework for making informed decisions under uncertainty.
• Statistical Inference: These are used to make inferences about populations based on sample data.

FAQs about the Probability Distribution of a Discrete Random Variable

1. Define the probability distribution.
2. What is a random variable?
3. What is meant by an expected value or a random variable?
4. What is meant by the variance and standard deviation of a random variable?

https://rfaqs.com, https://gmstat.com

We first need to understand the Bernoulli Trials to learn about Binomial Probability Distribution.

Bernoulli Trials

Many experiments consist of repeated independent trials and each trial has only two possible outcomes such as head or tail, right or wrong, alive or dead, defective or non-defective, etc. If the probability of each outcome remains the same (constant) throughout the trials, then such trials are called the Bernoulli Trials.

Binomial Probability Distribution

Binomial Probability Distribution is a discrete probability distribution describing the results of an experiment known as the Bernoulli Process. The experiment having $n$ Bernoulli trials is called a Binomial Probability experiment possessing the following four conditions/ assumptions

1. The experiment consists of $n$ repeated tasks.
2. Each trial results in an outcome that may be classified as success or failure.
3. The probability of success denoted by $p$ remains constant from trial to trial.
4. The repeated trials are independent.

A Binomial trial can result in a success with probability $p$ and a failure with probability $1-p$ having $n-x$ number of failures, then the probability distribution of Binomial Random Variable, the number of successes in $n$ independent trial is:

\begin{align*}
P(X=x)&=\binom{n}{x} \, p^x \, q^{n-x} \\
&=\frac{n!}{x!(n-x)!}\, p^x \, q^{n-x}
\end{align*}

The Binomial probability distribution is the most widely used in situations of two outcomes. It was discovered by the Swiss mathematician Jakob Bernoulli (1654—1704) whose main work on “the ars Conjectandi” (the art of conjecturing) was published posthumously in Basel in 1713.

Mean of Binomial Distribution:   Mean = $\mu = np$

Variance of Binomial Distribution:  Variance = $npq$

Standard Deviation of Binomial Distribution:  Standard Deviation = $\sqrt{npq}$

Moment Coefficient of Skewness:

\begin{align*}
\beta_1 &= \frac{q-p}{\sqrt{npq}}  \\
&= \frac{1-2p}{\sqrt{npq}}
\end{align*}

Moment Coefficient of Kurtosis:  $\beta_3 = 3+\frac{1-6pq}{npq}$

Application of Binomial Probability Distribution

• Quality control: In manufacturing, Binomial Probability Distribution can be used to determine the probability of finding a defective product in a batch.
• Medical testing: It can be used to assess the probability of a specific number of positive test results in a group.
• Opinion polls: Binomial Probability Distribution can be used to estimate the margin of error in a poll by considering the probability of getting a certain number of votes for a particular candidate.

By understanding the binomial distribution, you can analyze the probability of success in various scenarios with two possible outcomes.

FAQS about Binomial Probability Distribution

1. What is a Binomial Experiment?
2. Define Binomial Distribution?
3. What are the important Assumptions of a Binomial experiment?
4. What are the important applications of Binomial distribution?
5. What are the characteristics of Binomial distribution?
6. Write the probability distribution formula for a Binomial random variable.

Generate Binomial Random Numbers in R Language