In the world of probability and statistics, distributions are at the heart of everything we do. They describe how outcomes are spread out, help us model uncertainty, and form the foundation of decision-making in fields ranging from finance to artificial intelligence.

One of the first classes of probability distributions every student encounters is the discrete distribution. In this post, we’ll break down what it means, walk through some of the most important types, solve examples, and finish with practice exercises.

🔹 What is a Discrete Distribution?

A discrete distribution applies when a random variable can take only finite or countably infinite values.

Mathematically, if $X$ is a discrete random variable, then its probability mass function (PMF) is defined as: P(X=x)=p(x),where p(x)≥0and∑xp(x)=1P(X=x) = p(x), \quad \text{where } p(x) \geq 0 \quad \text{and} \quad \sum_x p(x) = 1P(X=x)=p(x),where p(x)≥0andx∑​p(x)=1

This means:

• Every outcome has a probability between 0 and 1.
• The sum of probabilities across all outcomes is always 1.

🔹 Key Discrete Distributions

1. Bernoulli Distribution

• The simplest discrete distribution.
• Describes a random experiment with only two possible outcomes: success (1) and failure (0).

P(X=x)={px=1,1−px=0P(X=x) = \begin{cases} p & x=1, \\ 1-p & x=0 \end{cases}P(X=x)={p1−p​x=1,x=0​

• Mean: $E[X] = p$
• Variance: $\text{Var}(X) = p(1-p)$

💡 Example: Tossing a coin once, where $p = 0.5$ for heads.

2. Binomial Distribution

• Models the number of successes in n independent Bernoulli trials.

P(X=k)=(nk)pk(1−p)n−k,k=0,1,…,nP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0,1,\dots,nP(X=k)=(kn​)pk(1−p)n−k,k=0,1,…,n

• Mean: $E[X] = np$
• Variance: $\text{Var}(X) = np(1-p)$

💡 Example: Number of heads in 10 tosses of a fair coin.

3. Poisson Distribution

• Models the number of events happening in a fixed time or space, assuming events occur independently and at a constant average rate $\lambda$.

P(X=k)=e−λλkk!,k=0,1,2,…P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}, \quad k=0,1,2,\ldotsP(X=k)=k!e−λλk​,k=0,1,2,…

• Mean = Variance = $\lambda$

💡 Example: Number of emails arriving per hour.

4. Geometric Distribution

• Describes the number of Bernoulli trials needed until the first success.

P(X=k)=(1−p)k−1p,k=1,2,3,…P(X=k) = (1-p)^{k-1}p, \quad k=1,2,3,\ldotsP(X=k)=(1−p)k−1p,k=1,2,3,…

• Mean: $E[X] = \frac{1}{p}$
• Variance: $\frac{1-p}{p^2}$

💡 Example: Probability that the first head appears on the 4th toss of a coin.

5. Discrete Uniform Distribution

• All outcomes are equally likely.

P(X=xi)=1n,i=1,2,…,nP(X=x_i) = \frac{1}{n}, \quad i=1,2,\ldots,nP(X=xi​)=n1​,i=1,2,…,n

💡 Example: Rolling a fair six-sided die.

🔹 Worked Examples

✅ Example 1: Binomial Distribution

A fair coin is tossed 4 times. Find the probability of getting exactly 2 heads.

• Here, $n = 4$, $p = 0.5$, $k = 2$.

P(X=2)=(42)(0.5)2(0.5)2=616=0.375P(X=2) = \binom{4}{2} (0.5)^2 (0.5)^2 = \frac{6}{16} = 0.375P(X=2)=(24​)(0.5)2(0.5)2=166​=0.375

So, there’s a 37.5% chance of exactly 2 heads.

✅ Example 2: Poisson Distribution

The number of cars arriving at a toll gate follows a Poisson distribution with mean $\lambda = 3$ per minute. Find the probability of exactly 2 cars arriving in a minute. P(X=2)=e−3⋅322!=9e−32≈0.224P(X=2) = \frac{e^{-3}\cdot 3^2}{2!} = \frac{9e^{-3}}{2} \approx 0.224P(X=2)=2!e−3⋅32​=29e−3​≈0.224

So, there’s about a 22.4% chance of exactly 2 cars arriving.

🔹 Practice Exercises

Try solving these on your own:

1. A die is rolled 10 times. What is the probability of getting exactly 3 sixes? (Binomial)
2. In a city, the average number of accidents per day is 2. What’s the probability of observing no accidents on a given day? (Poisson)
3. A student guesses randomly on 5 multiple-choice questions with 4 options each. What is the probability distribution of the number of correct answers? (Binomial)
4. A coin is tossed until the first head appears. What’s the probability that it takes exactly 4 tosses? (Geometric)
5. Write the PMF for rolling a fair die. (Uniform)

🔹 Summary

• Discrete distributions apply when the random variable takes finite or countable outcomes.
• Key types include Bernoulli, Binomial, Poisson, Geometric, and Uniform.
• Each distribution has specific formulas for probability, mean, and variance.
• They form the backbone of probability theory and real-world applications like quality control, risk modeling, and machine learning.

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