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Statistical Distributions
Edexcel A Level Mathematics – Statistics Revision Notes
Discrete Probability Distributions
A random variable is just a quantity whose value depends on the outcome of some random experiment. We use capital letters like X for random variables and small letters like x for specific values.
Basic Rules for Any Probability Distribution
Every probability must be between 0 and 1: 0 ≤ P (X = x) ≤ 1 All probabilities must add up to exactly 1:
P
P (X = x) = 1 These two rules let you find unknown values in probability tables.
Example: X has the distribution:
x 1 2 3 4 P (X = x) 0.1 p 0.3 0.
Since all probs sum to 1: 0.1 + p + 0.3 + 0.2 = 1, so p = 0.4.
The Binomial Distribution
This is the most important discrete distribution at A Level. Use it when you have a fixed number of independent trials, each with the same probability of success.
When to Use Binomial: The Four Conditions
- Fixed number of trials, n.
- Each trial results in either “success” or “failure.”
- Trials are independent – the outcome of one doesn’t affect another.
- The probability of success p is constant across all trials. If all four hold, write X ∼ B(n, p).
The Formula
P (X = r) =
n r
pr(1 − p)n−r
where
n r
n! r!(n − r)!
is the number of ways to choose r successes from n trials.
In practice, you use the binomial CD (cumulative distribution) on your calculator. Make sure you know how to find:
P (X = r) – exact value (use binomial PD on calculator)
P (X ≤ r) – cumulative (use binomial CD)
P (X ≥ r) = 1 − P (X ≤ r − 1) P (a ≤ X ≤ b) = P (X ≤ b) − P (X ≤ a − 1)
Mean and Variance of Binomial
E(X) = np Var(X) = np(1 − p) SD(X) =
p np(1 − p)
Quick Example
A biased coin has P (Heads) = 0.3. It is flipped 20 times. X = number of heads. X ∼ B(20, 0 .3) E(X) = 20 × 0 .3 = 6, Var(X) = 20 × 0. 3 × 0 .7 = 4. 2 P (X ≤ 5) = binomcdf(20, 0. 3 , 5) on your calculator. P (X ≥ 8) = 1 − P (X ≤ 7) = 1− binomcdf(20, 0. 3 , 7)
The Normal Distribution
The normal distribution is the classic “bell curve.” It’s used for continuous random variables
- things like heights, weights, or test scores. It’s symmetric about the mean.
Key Facts about Normal Distribution
If X ∼ N (μ, σ^2 ): μ is the mean (centre of the bell curve). σ^2 is the variance; σ is the standard deviation (how spread out it is). The distribution is perfectly symmetric about μ. P (X < μ) = P (X > μ) = 0. 5 About 68% of data lies within 1σ of the mean, 95% within 2σ, 99.7% within 3σ.
Standardising – The Z-Score
To use normal distribution tables, or to compare values from different normal distributions, you standardise by converting to a Z-score:
Z =
X − μ σ
where Z ∼ N (0, 1)
This tells you how many standard deviations a value is from the mean. On a calculator, you use normalcdf or invNorm directly without needing to standardise manually.
Finding Probabilities
P (X < a): use normalcdf(−∞, a, μ, σ) on calculator. P (X > a) = 1 − P (X < a)
P (a < X < b) = P (X < b) − P (X < a)
Symmetry shortcut: Because the distribution is symmetric, P (X > μ + k) = P (X < μ − k).
For normal distribution, always check whether the question gives variance or standard deviation.
