# Continuous Distributions (S2)

> Edexcel International A-Level Mathematics · IAL S2
> Source: https://www.owlsprep.com/study/edexcel-ial-math-s2-continuous-distributions/

This guide covers the two core continuous distribution topics for Edexcel IAL Maths S2: the continuous uniform distribution, and Normal approximations to binomial/Poisson distributions with continuity correction, aligned to the 2018 specification.

**Prerequisites:** [S1 Normal distribution properties](https://www.owlsprep.com/study/edexcel-ial-math-s1-normal-distributions/); [S2 Binomial and Poisson distributions](https://www.owlsprep.com/study/edexcel-ial-math-s2-discrete-probability-distributions/)

## Learning objectives

- Derive the cdf, mean and variance of the continuous uniform distribution
- Apply the continuous uniform distribution to solve probability problems
- Use Normal distribution to approximate binomial and Poisson distributions with correct continuity correction
- Recognise validity conditions for Normal approximations in exam questions

## 1. Continuous Uniform (Rectangular) Distribution

**Continuous Uniform Distribution** — 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.

*Notation:* $X \sim U(a,b)$

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

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

1. 1. CDF calculation: integrate the pdf from $a$ to $x$ for $a \leq x \leq b$:

   $$F(x) = \int_a^x \frac{1}{b-a} dt = \frac{x - a}{b - a}$$
2. 2. Mean calculation: use $E[X] = \int_a^b x f(x) dx$

   $$E[X] = \int_a^b \frac{x}{b-a} dx = \frac{1}{b-a} \left[ \frac{x^2}{2} \right]_a^b = \frac{b^2 - a^2}{2(b-a)} = \frac{a + b}{2}$$
3. 3. Variance calculation: use $Var(X) = E[X^2] - (E[X])^2$, first compute $E[X^2]$

   $$E[X^2] = \int_a^b \frac{x^2}{b-a} dx = \frac{1}{b-a} \left[ \frac{x^3}{3} \right]_a^b = \frac{b^3 - a^3}{3(b-a)} = \frac{a^2 + ab + b^2}{3}$$
4. Subtract $(E[X])^2 = \left(\frac{a+b}{2}\right)^2$:

   $$Var(X) = \frac{a^2 + ab + b^2}{3} - \frac{a^2 + 2ab + b^2}{4} = \frac{(b - a)^2}{12}$$

*Conclusion:* 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

1. Step 1: Define the random variable: Let $X$ = minutes past 10:00 the bus arrives, so $X \sim U(0, 15)$
2. Step 2: Calculate $P(5 < X < 12)$ using the interval length ratio:

   $$P(5 < X < 12) = \frac{12 - 5}{15 - 0} = \frac{7}{15} \approx 0.467$$
3. 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.

*Calculator:* allowed

## 2. Normal Approximation to the Binomial Distribution

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

*Notation:* $B(n,p) \approx N(np, np(1-p))$

> **warning**
>
> The binomial distribution is discrete, while the Normal distribution is continuous, so you *must apply a continuity correction* to all boundary values before calculating probabilities.

**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.

1. Step 1: Define the binomial random variable: Let $X$ = number of sixes rolled, so $X \sim B(120, 1/6)$
2. Step 2: Check approximation conditions: $np = 120 \times 1/6 = 20 >5$, $n(1-p)=100>5$, so approximation is valid
3. Step 3: State the Normal approximation parameters: mean $\mu = 20$, variance $\sigma^2 = 120 \times 1/6 \times 5/6 = 100/6 \approx 16.6667$, so $X \approx N(20, 16.6667)$
4. Step 4: Apply continuity correction for $P(X < 15)$: discrete $X <15$ corresponds to continuous $X < 14.5$
5. Step 5: Calculate the z-score:

   $$z = \frac{14.5 - 20}{\sqrt{16.6667}} = \frac{-5.5}{4.0825} \approx -1.347$$
6. 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.

*Calculator:* allowed

## 3. Normal Approximation to the Poisson Distribution

**Normal Approximation to Poisson** — When a Poisson distribution $X \sim Po(\lambda)$ has a large mean $\lambda > 10$, it can be approximated by a Normal distribution with matching mean and variance.

*Notation:* $Po(\lambda) \approx N(\lambda, \lambda)$

> **tip**
>
> For Poisson approximations, the variance equals the mean $\lambda$, so you only need to calculate one parameter to define the Normal distribution.

**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.

1. Step 1: Define the Poisson random variable: Let $X$ = number of customers per hour, so $X \sim Po(25)$
2. Step 2: Check approximation condition: $\lambda = 25 >10$, so approximation is valid
3. Step 3: State Normal approximation parameters: $\mu = 25$, $\sigma^2 = 25$, so $X \approx N(25,25)$
4. Step 4: Apply continuity correction for $P(X >30)$: discrete $X>30$ corresponds to continuous $X>30.5$
5. Step 5: Calculate z-score:

   $$z = \frac{30.5 -25}{\sqrt{25}} = \frac{5.5}{5} = 1.1$$
6. Step 6: Use Normal tables: $P(Z >1.1) = 1 - \Phi(1.1) = 1 - 0.8643 = 0.1357$, 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.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to derive the continuous uniform mean/variance when asked, just quoting the formula.
  - Why it fails: The specification explicitly requires you to demonstrate derivation from first principles, so full marks are not awarded for just stating the formula.
  - Correct: Show the full integration steps for E[X] and Var(X) whenever derivation is requested.
- **Wrong:** Applying Normal approximation to binomial when np or n(1-p) are less than 5.
  - Why it fails: The approximation is not valid for small sample sizes, leading to large errors in probability calculations.
  - Correct: Always check that both np and n(1-p) are >5 before using a Normal approximation for binomial.
- **Wrong:** Missing continuity correction when approximating discrete distributions.
  - Why it fails: 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: 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:** Using the standard deviation instead of variance in the Normal distribution notation.
  - Why it fails: Edexcel IAL uses $N(\mu, \sigma^2)$ notation, so writing $N(\mu, \sigma)$ will be marked as incorrect.
  - Correct: Always write the Normal parameters as (mean, variance) when stating the approximation.
- **Wrong:** Using continuity correction for continuous uniform distribution problems.
  - Why it fails: The continuous uniform distribution is already continuous, so no adjustment is needed for boundary values.
  - Correct: Only apply continuity correction when moving from a discrete (binomial/Poisson) to continuous (Normal) distribution.

## 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 | $B(n,p) \approx N(np, np(1-p))$ | $np$ | $np(1-p)$ | Valid if $np>5$ and $n(1-p)>5$, apply continuity correction |
| Normal approx to Poisson | $Po(\lambda) \approx N(\lambda, \lambda)$ | $\lambda$ | $\lambda$ | Valid if $\lambda>10$, apply continuity correction |

## 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.

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