Welcome to Statistical Distributions!

In this chapter, we are moving from looking at data we've already collected to modelling what might happen in the future. Think of a distribution as a "mathematical blueprint" that tells us how likely different outcomes are. Whether you are predicting how many heads you'll get in 100 coin flips or estimating the heights of people in a city, statistical distributions are your best friend.

Don't worry if this seems a bit abstract at first. We will break it down into two main types: Discrete (things you count) and Continuous (things you measure).

1. Discrete Random Variables

A Discrete Random Variable (usually written as \(X\)) is something that can only take specific, separate values. For example, the number of goals scored in a match can be 0, 1, or 2, but never 1.5!

The Discrete Uniform Distribution

This is the simplest distribution. It happens when every outcome is equally likely. Example: Rolling a fair six-sided die. Every number from 1 to 6 has a probability of exactly \(1/6\).

General Discrete Distributions

A distribution is often shown in a table where the total of all probabilities \(P(X=x)\) must equal 1. If they don't add up to 1, it’s not a valid distribution!

Quick Review:
  • Discrete: Countable values only.
  • Uniform: All outcomes have the same probability.
  • Sum of Probabilities: \(\sum P(X=x) = 1\).

2. The Binomial Distribution \(X \sim B(n, p)\)

The Binomial distribution is used when you have a fixed number of "trials" and you want to find the probability of a certain number of "successes."

When can we use it? (The BINS Criteria)

To use the Binomial model, four things must be true. You can remember them with the mnemonic BINS:

  • B - Binary: Only two possible outcomes (Success or Failure).
  • I - Independent: One trial doesn't affect the next.
  • N - Number: There is a fixed number of trials (\(n\)).
  • S - Success: The probability of success (\(p\)) stays the same for every trial.

Using your Calculator

For the 9MA0 syllabus, you don't need to use the big scary formula as much as your calculator! You need to know two modes:

  1. Binomial PD (Probability Density): Use this for "exactly" a value. E.g., \(P(X = 5)\).
  2. Binomial CD (Cumulative Distribution): Use this for "less than or equal to." E.g., \(P(X \leq 5)\).

Common Mistake: If the question asks for \(P(X > 5)\), remember that your calculator only does "less than or equal to." You must calculate \(1 - P(X \leq 5)\).

Key Takeaway: The Binomial distribution models the number of successes in \(n\) trials with a constant probability \(p\). Always check the BINS criteria before using it!

3. The Normal Distribution \(X \sim N(\mu, \sigma^2)\)

While the Binomial is for counting, the Normal Distribution is for measuring things like height, weight, or time. It is a continuous distribution, meaning \(X\) can take any value (like 1.752m).

The Bell Curve

The Normal distribution looks like a symmetrical bell shape:

  • The peak is at the Mean (\(\mu\)).
  • It is perfectly symmetrical around the mean.
  • The total area under the curve is 1 (representing 100% probability).
  • The "spread" is determined by the Standard Deviation (\(\sigma\)).

Points of Inflection

Did you know? The "curve" of the bell changes from bending downwards to bending upwards at exactly one standard deviation away from the mean. Important point: The points of inflection are at \(x = \mu + \sigma\) and \(x = \mu - \sigma\).

Finding Probabilities

Use the Normal CD mode on your calculator. You will need to input a Lower bound, an Upper bound, \(\sigma\), and \(\mu\). Example: To find \(P(X > 10)\), set your lower bound to 10 and your upper bound to a very large number like 99,999.

Quick Review:
  • \(\mu\): The center of the bell.
  • \(\sigma\): How wide the bell is.
  • Total area: Always equals 1.

4. Normal Approximation to the Binomial

Sometimes, a Binomial distribution is so large that it starts looking just like a Normal distribution. We can use the Normal distribution to approximate a Binomial distribution to make calculations easier.

When can we do this?

You can approximate \(B(n, p)\) with a Normal distribution only if:

  • \(n\) is large (usually \(n > 50\)).
  • \(p\) is close to 0.5 (the distribution is not too lopsided).

The Parameters

If \(X \sim B(n, p)\), then the approximating Normal distribution \(Y \sim N(\mu, \sigma^2)\) uses:

  • Mean: \(\mu = np\)
  • Variance: \(\sigma^2 = np(1 - p)\)

The Continuity Correction

This is the part that trips up most students! Because we are moving from a discrete "staircase" (Binomial) to a smooth "slope" (Normal), we have to adjust the numbers slightly.

  • If Binomial asks for \(P(X \leq 5)\), use Normal \(P(Y < 5.5)\).
  • If Binomial asks for \(P(X \geq 5)\), use Normal \(P(Y > 4.5)\).
  • If Binomial asks for \(P(X = 5)\), use Normal \(P(4.5 < Y < 5.5)\).

Think of it this way: To capture the whole "bar" for the number 5, you need to go from 4.5 to 5.5.

5. Choosing and Critiquing Distributions

In your exam, you might be asked why a model is not appropriate. Here are some common reasons:

Why Binomial might fail:

  • The trials are not independent (e.g., picking socks from a drawer without replacement).
  • The probability changes (e.g., weather patterns).

Why Normal might fail:

  • The data is skewed (not symmetrical).
  • The data has "extreme outliers" that the Normal model doesn't account for.
Key Takeaway: Real life is messy! Models are just simplifications. Always look for words like "independent" or "random" to justify using these distributions.

Final Exam Tips for 9MA0

  • Read the Notation: \(N(\mu, \sigma^2)\) uses variance. If the question says \(N(10, 16)\), then \(\sigma = 4\). Don't forget to square root it for your calculator!
  • Sketch the Curve: For Normal distribution questions, always draw a quick bell curve and shade the area you are looking for. It helps prevent silly mistakes with Upper/Lower bounds.
  • Simultaneous Equations: If you need to find \(\mu\) and \(\sigma\), use the percentage points table or the Inverse Normal function on your calculator to set up two equations.
  • Check your bounds: For Binomial CD, always check if the question is \(\leq\) or \(<\). Your calculator only does \(\leq\).

You've got this! Statistical distributions are just a set of tools. Once you know which tool to pick and which buttons to press on your calculator, the marks will follow!