Distribution of Discrete Random Variables in Statistics

A discrete random variable is a type of random variable that can take on a countable number of distinct values. The probability distribution of a discrete random variable describes the probabilities associated with each possible value the variable can assume. These distributions are essential in various fields such as economics, engineering, and natural sciences for modeling and analyzing data that can only take on specific values.

Key Concepts

Probability Mass Function (PMF):

• The PMF of a discrete random variable $X$ is a function $P(X = x)$ that gives the probability that $X$ takes the value $x$ .
• The PMF satisfies two key properties:
  1. $P(X = x) \geq 0$ for all $x$ .
  2. $\sum_x P(X = x) = 1$ .

Cumulative Distribution Function (CDF):

• The CDF of a discrete random variable $X$ , denoted as $F(x)$ , is the probability that $X$ takes a value less than or equal to $x$ : $F(x) = P(X \leq x) = \sum_{t \leq x} P(X = t)$
• The CDF is a non-decreasing function that approaches 0 as $x$ approaches $-\infty$ and 1 as $x$ approaches $+\infty$ .

Common Discrete Distributions

1. Bernoulli Distribution:

• A Bernoulli random variable takes the value 1 with probability $p$ and the value 0 with probability $1 - p$ .
• PMF: p & \text{if } x = 1 \\ 1 - p & \text{if } x = 0 \end{cases} \]
• Example: The outcome of a single coin toss where heads is coded as 1 and tails as 0.

2. Binomial Distribution:

A Binomial random variable represents the number of successes in $n$ independent Bernoulli trials, each with probability $p$ of success.

PMF: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ where $k$ is the number of successes, and $\binom{n}{k}$ is the binomial coefficient.

Example: The number of heads in 10 tosses of a fair coin.

3. Geometric Distribution:

A Geometric random variable represents the number of trials needed to get the first success in a sequence of independent Bernoulli trials.

PMF: $P(X = k) = (1 - p)^{k - 1} p$ where $k$ is the number of trials until the first success.

Example: The number of times a die is rolled until a 6 appears.

4. Poisson Distribution:

A Poisson random variable represents the number of events occurring in a fixed interval of time or space, given the events occur with a constant mean rate $\lambda$ and independently of the time since the last event.

PMF: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$ where $k$ is the number of events.

Example: The number of emails received in an hour.

Examples with Numerical Calculations

Example 1: Binomial Distribution

A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

• Let $X$ be the number of heads.
• $X$ follows a Binomial distribution with $n = 5$ and $p = 0.5$ .

Using the Binomial PMF: $P(X = 3) = \binom{5}{3} (0.5)^3 (0.5)^{5 - 3}$ $P(X = 3) = \frac{5!}{3!(5 - 3)!} (0.5)^3 (0.5)^2$ $P(X = 3) = \frac{5 \times 4}{2 \times 1} \times 0.125 \times 0.25$ $P(X = 3) = 10 \times 0.125 \times 0.25$ $P(X = 3) = 0.3125$

So, the probability of getting exactly 3 heads in 5 tosses of a fair coin is 0.3125.

Example 2: Poisson Distribution

A call center receives an average of 3 calls per minute. What is the probability that exactly 5 calls are received in a given minute?

• Let $X$ be the number of calls received in a minute.
• $X$ follows a Poisson distribution with $\lambda = 3$ .

Using the Poisson PMF: $P(X = 5) = \frac{3^5 e^{-3}}{5!}$ $P(X = 5) = \frac{243 \times e^{-3}}{120}$ $P(X = 5) \approx \frac{243 \times 0.0498}{120}$ $P(X = 5) \approx 0.1008$

So, the probability of receiving exactly 5 calls in a minute is approximately 0.1008.

Applications

1. Quality Control:

• Discrete distributions are used to model the number of defective items in a batch.
• Example: Using the Binomial distribution to determine the probability of finding a certain number of defective items in a sample.

2. Risk Assessment in Insurance:

• Poisson distribution is used to model the number of claims received in a given period.
• Example: Calculating the probability of receiving a specific number of claims in a month.

3. Inventory Management:

• Geometric distribution helps in modeling the number of trials until a restock is needed.
• Example: Determining the probability of the next restock occurring after a certain number of sales.

4. Telecommunications:

• Poisson distribution is used to model the number of incoming calls or messages.
• Example: Estimating the probability of receiving a certain number of messages in an hour.

Conclusion

Understanding the distribution of discrete random variables is essential in statistics for analyzing and interpreting data that take on distinct values. By using PMFs, CDFs, and specific distributions like Binomial, Geometric, and Poisson, statisticians can model various real-world scenarios, calculate probabilities, and make informed decisions based on their analyses. These distributions are widely applicable across different fields, making them invaluable tools in statistical practice.