Navigate the Probability Field
S2 develops the statistical systems introduced earlier, moving from discrete probability models into continuous distributions, normal approximations and formal hypothesis testing.
FIELD 01 • DISCRETE DISTRIBUTIONS
The Binomial and Poisson Distributions
S2 begins by developing the probability distributions introduced in S1. The Binomial and Poisson distributions can be used to model different types of discrete random variables.
Binomial Distribution
A random variable X has a Binomial distribution when there is a fixed number of independent trials, each trial has two possible outcomes, and the probability of success remains constant.
Conditions for a Binomial Model
- There is a fixed number of trials, n.
- Each trial has two possible outcomes.
- The probability of success, p, is constant.
- The trials are independent.
Binomial Probability
Mean and Variance
Modelling with the Binomial Distribution
A model should only be used when its assumptions are reasonable. In examination questions, you may need to explain whether a Binomial model is appropriate.
Worked Example
A machine produces components. The probability that a component is defective is 0.08. A random sample of 20 components is selected. Let X be the number of defective components.
Each component has two possible outcomes: defective or non-defective.
The probability of a defective component remains constant at 0.08.
Assuming the components are independent:
X ~ B(20, 0.08)
Mean = 20 × 0.08 = 1.6
Variance = 20 × 0.08 × 0.92 = 1.472
Poisson Distribution
The Poisson distribution can be used to model the number of times an event occurs in a fixed interval of time, length, area or another suitable interval.
Poisson Probability
Mean and Variance
For a Poisson distribution, the mean and variance are equal.
Additive Property
If two independent Poisson random variables are added, the result is also Poisson.
Poisson Approximation to the Binomial
Under suitable conditions, a Binomial distribution can be approximated by a Poisson distribution.
The approximation is particularly useful when n is large and p is small.
Always consider whether the assumptions of the model are reasonable before using an approximation.
FIELD 02 • CONTINUOUS VARIABLES
Continuous Random Variables
Unlike a discrete random variable, a continuous random variable can take any value within an interval.
Probability Density Function
A probability density function, or PDF, describes how probability is distributed over the possible values of X.
- f(x) ≥ 0 for all possible values of x.
- The total area under the curve is 1.
Finding Probabilities
For a continuous random variable, probability is represented by area under the probability density function.
Probability at a Single Point
Cumulative Distribution Function
The cumulative distribution function gives the probability that X is less than or equal to a particular value.
Relationship Between PDF and CDF
Worked Example
A continuous random variable X has density f(x) = 2x for 0 ≤ x ≤ 1. Find P(X ≤ 0.5).
P(X ≤ 0.5) = ∫₀⁰·⁵ 2x dx
= [x²]₀⁰·⁵
= 0.25
P(X ≤ 0.5) = 0.25
Simple Polynomial Density Functions
S2 questions may involve probability density functions defined by simple polynomial expressions over a specified interval.
You may need to use the condition that the total probability is 1 to find an unknown constant.
FIELD 03 • DISTRIBUTION LANDSCAPE
Continuous Distributions
Mean of a Continuous Random Variable
The mean of a continuous random variable is found by weighting each possible value by its probability density.
Second Moment
Variance
Standard Deviation
Mode
The mode is the value of X at which the probability density function reaches its maximum.
Median
The median is the value m for which half of the probability lies below m and half lies above m.
Quartiles
Quartiles divide the distribution into four equal probability sections.
Worked Example
Suppose the cumulative distribution function is known. To find the median, set the cumulative probability equal to 0.5 and solve for x.
Set:
F(x) = 0.5
Solve the resulting equation for x.
The solution is the median.
Continuous Uniform Distribution
A continuous uniform distribution has constant probability density throughout a specified interval.
Probability Density
Mean
Variance
The uniform distribution gives equal probability density across its entire interval.
FIELD 04 • NORMAL TRANSFORMATION
Normal Approximations
The Normal distribution can be used to approximate certain Binomial and Poisson distributions when appropriate.
Normal Distribution
Binomial to Normal
If X ~ B(n,p), the approximating Normal distribution has:
Poisson to Normal
If X ~ Po(λ), a suitable Normal approximation has:
Continuity Correction
A Binomial or Poisson random variable is discrete, whereas the Normal distribution is continuous. A continuity correction adjusts the boundary by 0.5.
P(X ≥ 10) becomes P(Y > 9.5).
P(X ≤ 20) becomes P(Y < 20.5).
P(10 ≤ X ≤ 20) becomes P(9.5 < Y < 20.5).
Standardisation
Worked Example
X ~ B(100, 0.4). Approximate P(X ≥ 45) using a Normal distribution.
Mean:
μ = np = 100 × 0.4 = 40
Variance:
σ² = np(1 − p) = 100 × 0.4 × 0.6 = 24
Therefore use Y ~ N(40, 24).
Apply continuity correction:
P(X ≥ 45) ≈ P(Y > 44.5)
Then standardise using:
Z = (44.5 − 40) / √24
FIELD 05 • INFERENCE SIGNAL
Hypothesis Tests
Hypothesis testing provides a formal method for deciding whether sample evidence is sufficiently strong to reject a statistical claim.
Null Hypothesis
The null hypothesis, H₀, represents the claim or assumption being tested.
Alternative Hypothesis
The alternative hypothesis, H₁, represents the competing claim.
Significance Level
The significance level is the probability of rejecting H₀ when H₀ is actually true.
Critical Region
The critical region contains values of the test statistic that would lead to rejection of H₀.
Binomial Hypothesis Tests
Binomial models can be used to test claims about the probability of success p.
Worked Example Structure
Step 1: State H₀ and H₁.
Step 2: Identify the distribution under H₀.
Step 3: Find the relevant probability or critical value.
Step 4: Compare the result with the significance level.
Step 5: State the conclusion in context.
Poisson Hypothesis Tests
A Poisson distribution can be used to test claims about the mean rate λ.
Testing Procedure
- Define the null and alternative hypotheses.
- Use the null hypothesis to specify the distribution.
- Identify the critical region or calculate a probability.
- Compare the evidence with the significance level.
- Reject or do not reject H₀.
- State the conclusion in the context of the question.
Do not simply say “H₀ is true” when a test result is not significant. Say that there is insufficient evidence to reject H₀.
Normal Approximation in Hypothesis Testing
When appropriate, a Normal approximation can be used for a Binomial hypothesis test. The continuity correction must be applied when making the approximation.
FIELD TEST • MIXED PRACTICE
S2 Mixed Practice
These questions combine the major S2 techniques rather than separating practice by topic.
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Fixed number of trials, two possible outcomes per trial, constant probability of success and independent trials.
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E(X) = 6 and Var(X) = 6.
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When n is sufficiently large and p is sufficiently small, using λ = np.
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The total area must equal 1.
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P(X = 4) = 0.
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F(x) is obtained by integrating f(x), and f(x) is the derivative of F(x).
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E(X) = (2 + 10) / 2 = 6.
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Mean = np = 20.
Variance = np(1 − p) = 100 × 0.2 × 0.8 = 16.
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Use P(Y > 29.5).
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There is insufficient evidence to reject H₀.
FIELD MEMORY CORE • QUICK REVIEW
S2 Key Formulae
- Check the assumptions of a probability model.
- Use the correct distribution.
- Remember continuity correction for Normal approximations.
- Use integration correctly for continuous variables.
- State hypothesis-test conclusions in context.
THE PROBABILITY FIELD
Every observation enters as uncertainty. The field transforms uncertainty into measurable probability, distributions and statistical evidence.