Common Probability Distributions

What is a Probability Distribution?

A probability distribution is a mathematical function that describes the likelihood of different possible outcomes for a random variable. To get a clear picture of the fundamentals, including the distinction between discrete and continuous variables, please watch the first part of this video.

To summarize the key points from the video:

• Discrete Distributions: Model random variables that can take on a finite or countable number of distinct values. Think of counting the number of heads in three coin flips (0, 1, 2, or 3). We use a Probability Mass Function (PMF) to describe the probability of each specific outcome.
• Continuous Distributions: Model random variables that can take any value within a given range. Think of measuring a person's height. We use a Probability Density Function (PDF), where the probability is represented by the area under the curve over an interval.

Now, let's dive into the specific distributions you need to know for the exam.

Discrete Distributions: Binomial and Poisson

You mentioned you've encountered these before, so we'll treat this as a quick refresher focused on their core applications and properties. The next segment of the video we just started explains the Bernoulli, Binomial, and Poisson distributions.

Key Characteristics

Here’s a breakdown of what you just reviewed:

1. Binomial Distribution

   • What it models: The number of "successes" in a fixed number (n) of independent trials, where each trial has only two possible outcomes (e.g., success/failure, yes/no, heads/tails).
   • Parameters:
     • $n$: The number of trials.
     • $p$: The probability of success on a single trial.
   • Formula (PMF): $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$
   • Key Idea: Think of it as repeating a simple two-outcome experiment (a Bernoulli trial) n times and counting the successes.
   • Example: Calculating the probability of getting exactly 7 heads in 10 coin flips.

2. Poisson Distribution

   • What it models: The number of events occurring in a fixed interval of time or space, given a known average rate of occurrence.
   • Parameter:
     • $\lambda$ (lambda): The average number of events per interval.
   • Formula (PMF): $P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}$
   • Key Idea: Use this when you are counting occurrences of an event over a continuous interval. The events must be independent.
   • Example: Modeling the number of emails you receive in an hour, given you receive an average of 5 emails per hour.

A common point of confusion is the relationship between the Poisson distribution and the Exponential distribution, which we'll cover next. This short video offers excellent intuition on how they, and the underlying Poisson process, are connected.

The crucial takeaway is:

• Poisson asks: How many events happen in a fixed time interval? (Discrete)
• Exponential asks: How much time passes between events? (Continuous)

Continuous Distributions: Normal and Exponential

Now let's turn to continuous distributions. We'll use the same video resource to get a quick overview of the Normal and Exponential distributions.

Key Characteristics

1. Normal Distribution (Gaussian Distribution)

   • What it models: Many natural phenomena like height, blood pressure, measurement errors, etc. It is the most important distribution in statistics.
   • Parameters:
     • $\mu$ (mu): The mean, which is the center of the distribution.
     • ${\sigma }^2$ (sigma-squared): The variance, which determines the spread of the curve.
   • Key Idea: It's a symmetric, bell-shaped curve. The 68-95-99.7 rule is a useful heuristic: about 68% of data falls within 1 standard deviation ($\sigma$) of the mean, 95% within 2$\sigma$, and 99.7% within 3$\sigma$.
   • Example: The distribution of IQ scores in a population, which has a mean of 100 and a standard deviation of 15.

2. Exponential Distribution

   • What it models: The time between events in a Poisson process. It describes the waiting time for an event to occur.
   • Parameter:
     • $\lambda$ (lambda): The rate parameter (the same $\lambda$ as in the related Poisson distribution).
   • Key Idea: The probability is highest at the beginning and decreases exponentially over time. It is "memoryless," meaning the probability of an event occurring in the future is independent of how much time has already passed.
   • Example: Modeling the amount of time until a customer arrives at a store.

The image below provides a great visual comparison of the shapes of these distributions.

Summary and Quick Reference

To meet your goal of having quick refresher notes, the following PDF from GeeksforGeeks is an excellent resource. It provides all the key formulas and properties in a very condensed format, perfect for last-minute revision.

This image also serves as a fantastic one-page cheat sheet.

Conclusion

In this lesson, we completed a rapid review of four essential probability distributions.

Key Takeaways:

• Binomial Distribution: Counts successes in a fixed number of binary trials.
• Poisson Distribution: Counts event occurrences in a fixed interval.
• Exponential Distribution: Measures the time between events.
• Normal Distribution: A symmetric, bell-shaped curve that models a vast range of natural and social phenomena.

Understanding the "story" behind each distribution—what it models and why—is the most critical part for your exam.

Preview of the Next Lesson:
In our next lesson, we will continue our journey through statistics by introducing two more distributions that are crucial for hypothesis testing: the t-distribution and the Chi-squared distribution. Understanding these will pave the way for the statistical inference topics that follow.