Introduction to Statistical Distributions
Welcome! In this chapter, we are going to look at how we can model real-life events using mathematics. A statistical distribution is essentially a "map" or a list that shows all the possible outcomes of an experiment and how likely each one is to happen. Whether you are predicting the number of heads in a coin flip or how many seeds in a packet will grow, these tools help us make sense of the world using probability. Don't worry if this seems a bit abstract at first; we'll use plenty of examples to keep it grounded!
1. Discrete Random Variables
Before we dive into the specific distributions, we need to understand what we are measuring. We use something called a Discrete Random Variable (usually written as \( X \)).
● Random: The outcome is due to chance.
● Variable: It can take different values.
● Discrete: It can only take specific, separate values (like 1, 2, 3...) rather than any value on a scale (like 1.543...).
Example: If you roll a die, the number it lands on is a discrete random variable because you can get a 1 or a 2, but you can't get a 1.5.
2. The Discrete Uniform Distribution
This is the simplest type of distribution. In a Discrete Uniform Distribution, every possible outcome has exactly the same probability of occurring.
The Classic Example: A Fair Die
If you roll a fair six-sided die, the possible outcomes are {1, 2, 3, 4, 5, 6}.
The probability of getting any one of these numbers is exactly \( \frac{1}{6} \). Because the probabilities are all equal (uniform), we call this a discrete uniform distribution.
Quick Review Box
● Discrete: You can count the outcomes.
● Uniform: All outcomes have the same chance.
3. The Binomial Distribution
The Binomial Distribution is the "superstar" of this chapter. It is used to model the number of "successes" in a fixed number of trials. For example, "If I throw 10 basketball free-throws, what is the probability I make exactly 7?"
When can we use a Binomial Model?
To use this model, the situation must meet four strict criteria. You can remember them using the mnemonic BINS:
● B - Binary: There are only two possible outcomes for each trial (usually called Success and Failure).
● I - Independent: The outcome of one trial does not affect the next one.
● N - Number: There is a fixed number of trials (we call this \( n \)).
● S - Success: The probability of success (we call this \( p \)) must be the same for every trial.
Notation
If a variable \( X \) follows a binomial distribution, we write it like this:
\( X \sim B(n, p) \)
● \( n \) = number of trials
● \( p \) = probability of success
Example: If you flip a fair coin 10 times and count the number of heads, you would write: \( X \sim B(10, 0.5) \).
4. Calculating Binomial Probabilities
In your exam, you are expected to use your calculator to find these probabilities. There are two main types of calculations you will do:
A. Probability Density (PD)
Use this when you want to find the probability of an exact value.
\( P(X = r) \)
Example: What is the probability of getting exactly 3 heads in 10 flips?
B. Cumulative Distribution (CD)
Use this when you want to find the probability of a range of values. Most calculators find \( P(X \le r) \), which means "r or fewer".
Example: What is the probability of getting at most 3 heads? This would be \( P(X \le 3) \), which is the same as \( P(X=0) + P(X=1) + P(X=2) + P(X=3) \).
Common Mistake: "More than" vs "At least"
Calculators usually only do "less than or equal to" (\( \le \)). If a question asks for \( P(X \ge 3) \) (at least 3), you must use the complement rule:
\( P(X \ge 3) = 1 - P(X \le 2) \)
Top Tip: Always write out the list of possible numbers (0, 1, 2, 3, 4, 5...) and circle the ones you want. This makes it much easier to see what you need to subtract from 1!
5. Expected Value (Mean)
While you don't need to calculate complex variance for AS Level, you should have an "informal appreciation" of the expected value. This is simply the average number of successes you would expect if you ran the experiment many times.
The formula is: Expected Value = \( n \times p \)
Example: If a seed has a 0.8 chance of growing and you plant 100 seeds, you would "expect" \( 100 \times 0.8 = 80 \) seeds to grow.
6. Modeling Real-World Situations
You may be asked to comment on whether a Binomial Distribution is a good model for a real-life situation. Look for reasons why the BINS criteria might fail.
Example: Picking marbles from a bag without replacing them.
● Is it Binomial? No.
● Why? Because if you don't put the marble back, the probability of picking a certain color changes for the next turn. This violates the "S" (Same probability) and "I" (Independence) in BINS.
Did you know?
The Binomial Distribution was named after the "Binomial Expansion" you study in Pure Maths! The coefficients (the numbers from Pascal's triangle) are the same ones used to calculate these probabilities.
Summary: Key Takeaways
● Use Discrete Uniform when all outcomes are equally likely (like a fair die).
● Use Binomial \( B(n, p) \) when you have a fixed number of trials with only two outcomes.
● Always check BINS before assuming a situation is Binomial.
● Master your calculator's "Binomial PD" and "Binomial CD" functions—they are your best friends in the exam!
● For \( P(X \ge r) \), remember to do \( 1 - P(X \le r-1) \).