Continuous Distributions (S2)

Q: When do I use continuity correction for Normal approximations?

Use continuity correction every time you approximate a discrete distribution (binomial or Poisson) with the continuous Normal distribution, adjusting the discrete boundary by ±0.5 to match the continuous scale.

Edexcel International A-Level Mathematics· 2018 Issue 3 S2 §3.1, §3.2· 25 min read

1. 1. Continuous Uniform (Rectangular) Distribution★★☆☆☆⏱ 8 min

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📘 Definition

Continuous Uniform Distribution

$X \sim U (a, b)$

A continuous distribution where all values in the interval $[a, b]$ are equally likely to occur, with constant pdf $f (x) = \frac{1}{b - a}$ for $a \leq x \leq b$ , 0 otherwise.

🔬 Derivation

Goal:

Derive the cdf, mean and variance of $X \sim U (a, b)$

Starting from:

pdf $f (x) = \frac{1}{b - a}$ for $a \leq x \leq b$

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1. CDF calculation: integrate the pdf from $a$ to $x$ for $a \leq x \leq b$ :
$F(x) = \int_a^x \frac{1}{b-a} dt = \frac{x - a}{b - a}$
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1. Mean calculation: use $E [X] = \int_{a}^{b} x f (x) d x$
$E[X] = \int_a^b \frac{x}{b-a} dx = \frac{1}{b-a} \left[ \frac{x^2}{2} \right]_a^b = \frac{b^2 - a^2}{2(b-a)} = \frac{a + b}{2}$
3
1. Variance calculation: use $V a r (X) = E [X^{2}] - (E [X])^{2}$ , first compute $E [X^{2}]$
$E[X^2] = \int_a^b \frac{x^2}{b-a} dx = \frac{1}{b-a} \left[ \frac{x^3}{3} \right]_a^b = \frac{b^3 - a^3}{3(b-a)} = \frac{a^2 + ab + b^2}{3}$
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Subtract $(E [X])^{2} = (\frac{a + b}{2})^{2}$ :
$Var(X) = \frac{a^2 + ab + b^2}{3} - \frac{a^2 + 2ab + b^2}{4} = \frac{(b - a)^2}{12}$

Result:

The cdf of $U (a, b)$ is $F (x) = \frac{x - a}{b - a}$ for $a \leq x \leq b$ , with mean $\frac{a + b}{2}$ and variance $\frac{( b - a ) ^{2}}{12}$ .

📐 Worked Example

A bus arrives at a stop uniformly between 10:00 and 10:15. Find the probability the bus arrives between 10:05 and 10:12, and the expected

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Step 1: Define the random variable: Let $X$ = minutes past 10:00 the bus arrives, so $X \sim U (0, 15)$
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Step 2: Calculate $P (5 < X < 12)$ using the interval length ratio:
$P(5 < X < 12) = \frac{12 - 5}{15 - 0} = \frac{7}{15} \approx 0.467$
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Step 3: Expected
$E[X] = \frac{0 + 15}{2} = 7.5 \text{ minutes}$

Exam tip:

You may be asked to derive the uniform mean/variance explicitly in exams, so memorise the integration steps even though the final formulae are given in the formula booklet.

2. 2. Normal Approximation to the Binomial Distribution★★★☆☆⏱ 8 min

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📘 Definition

Normal Approximation to Binomial

$B (n, p) \approx N (n p, n p (1 - p))$

When a binomial distribution $X \sim B (n, p)$ has large $n$ and $n p$ and $n (1 - p)$ are both greater than 5, it can be approximated by a Normal distribution with matching mean and variance.

📐 Worked Example

A fair six-sided die is rolled 120 times. Use a Normal approximation to find the probability of rolling fewer than 15 sixes.

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Step 1: Define the binomial random variable: Let $X$ = number of sixes rolled, so $X \sim B (120, 1/6)$
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Step 2: Check approximation conditions: $n p = 120 \times 1/6 = 20 > 5$ , $n (1 - p) = 100 > 5$ , so approximation is valid
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Step 3: State the Normal approximation parameters: mean $μ = 20$ , variance $σ^{2} = 120 \times 1/6 \times 5/6 = 100/6 \approx 16.6667$ , so $X \approx N (20, 16.6667)$
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Step 4: Apply continuity correction for $P (X < 15)$ : discrete $X < 15$ corresponds to continuous $X < 14.5$
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Step 5: Calculate the z-score:
$z = \frac{14.5 - 20}{\sqrt{16.6667}} = \frac{-5.5}{4.0825} \approx -1.347$
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Step 6: Use Normal tables to find the probability: $P (Z < - 1.347) \approx 0.0889$ , rounded to 3 significant figures as required.

Exam tip:

Edexcel exam markers will penalise missing continuity correction heavily, so always double check you have adjusted discrete boundaries by ±0.5 before computing z-scores.

3. 3. Normal Approximation to the Poisson Distribution★★★☆☆⏱ 7 min

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📘 Definition

Normal Approximation to Poisson

$P o (λ) \approx N (λ, λ)$

When a Poisson distribution $X \sim P o (λ)$ has a large mean $λ > 10$ , it can be approximated by a Normal distribution with matching mean and variance.

📐 Worked Example

The number of customers entering a café per hour follows a Poisson distribution with mean 25. Use a Normal approximation to find the probability that more than 30 customers enter in one hour.

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Step 1: Define the Poisson random variable: Let $X$ = number of customers per hour, so $X \sim P o (25)$
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Step 2: Check approximation condition: $λ = 25 > 10$ , so approximation is valid
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Step 3: State Normal approximation parameters: $μ = 25$ , $σ^{2} = 25$ , so $X \approx N (25, 25)$
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Step 4: Apply continuity correction for $P (X > 30)$ : discrete $X > 30$ corresponds to continuous $X > 30.5$
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Step 5: Calculate z-score:
$z = \frac{30.5 -25}{\sqrt{25}} = \frac{5.5}{5} = 1.1$
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Step 6: Use Normal tables: $P (Z > 1.1) = 1 - Φ (1.1) = 1 - 0.8643 = 0.1357$ , rounded to 0.136 (3 s.f.)

Exam tip:

If the question asks you to use a Normal approximation, you do not need to calculate exact Poisson/binomial probabilities for comparison, unless explicitly instructed.

4. Common Pitfalls

Wrong move:

Forgetting to derive the continuous uniform mean/variance when asked, just quoting the formula.

Why:

The specification explicitly requires you to demonstrate derivation from first principles, so full marks are not awarded for just stating the formula.

Correct move:

Show the full integration steps for E[X] and Var(X) whenever derivation is requested.

Wrong move:

Applying Normal approximation to binomial when np or n(1-p) are less than 5.

Why:

The approximation is not valid for small sample sizes, leading to large errors in probability calculations.

Correct move:

Always check that both np and n(1-p) are >5 before using a Normal approximation for binomial.

Wrong move:

Missing continuity correction when approximating discrete distributions.

Why:

The Normal distribution is continuous, so discrete boundary values need adjustment to align with the continuous scale, missing this leads to incorrect probability values and lost marks.

Correct move:

Adjust all discrete boundaries by ±0.5: e.g. $P (X \leq 10)$ becomes $P (X \leq 10.5)$ , $P (X > 7)$ becomes $P (X > 7.5)$ .

Wrong move:

Using the standard deviation instead of variance in the Normal distribution notation.

Why:

Edexcel IAL uses $N (μ, σ^{2})$ notation, so writing $N (μ, σ)$ will be marked as incorrect.

Correct move:

Always write the Normal parameters as (mean, variance) when stating the approximation.

Wrong move:

Using continuity correction for continuous uniform distribution problems.

Why:

The continuous uniform distribution is already continuous, so no adjustment is needed for boundary values.

Correct move:

Only apply continuity correction when moving from a discrete (binomial/Poisson) to continuous (Normal) distribution.

5. Quick Reference Cheatsheet

Distribution	Notation	Mean	Variance	Key Rules
Continuous Uniform	$U (a, b)$	$\frac{a + b}{2}$	$\frac{( b - a ) ^{2}}{12}$	Derive cdf/mean/variance via integration, no continuity correction needed
Normal approx to Binomial	$B (n, p) \approx N (n p, n p (1 - p))$	$n p$	$n p (1 - p)$	Valid if $n p > 5$ and $n (1 - p) > 5$ , apply continuity correction
Normal approx to Poisson	$P o (λ) \approx N (λ, λ)$	$λ$	$λ$	Valid if $λ > 10$ , apply continuity correction

6. Frequently Asked

When do I use continuity correction for Normal approximations?

Use continuity correction every time you approximate a discrete distribution (binomial or Poisson) with the continuous Normal distribution, adjusting the discrete boundary by ±0.5 to match the continuous scale.

Do I need to memorise the continuous uniform mean and variance formulae?

These are given in the official formula booklet, but you must be able to derive them from first principles as required by the S2 specification.

Going deeper

formula_bookEdexcel IAL Maths Formulae & Statistical TablesProvided in all S2 exams

What's Next

Now that you have mastered continuous distributions for S2, you can move on to hypothesis testing for discrete and continuous distributions, which forms the next major topic in the Edexcel IAL S2 specification. You will also apply these Normal approximation skills in later hypothesis testing questions, so make sure you are confident with continuity correction and parameter matching before progressing. If you need to revise prerequisite topics, revisit S1 Normal distributions or S2 discrete probability distributions first.