Theoretical Distributions for CA Foundation
Theoretical distributions: binomial, Poisson and normal, their means, variances and key properties.
Quantitative AptitudePaper 3, Statistics12 MCQs2 lessons2 flashcards3 memory tricks
Practise Theoretical Distributions free
What you'll learn
- Binomial and Poisson distributions
Compute means, variances and probabilities for binomial and Poisson distributions. - Normal distribution
Use the properties of the normal curve and z-scores.
Key formulas
| z-score | z = (x − μ)/σ |
|---|---|
| Standard normal | Mean 0, SD 1 |
Concepts explained
Binomial distribution
The binomial distribution gives the probability of x successes in n independent trials with success probability p: P(x) = ⁿCₓ pˣ qⁿ⁻ˣ.
What it means
- Mean = np; variance = npq; SD = √(npq).
- Mean > variance because q < 1.
- Symmetric when p = 0.5.
- From mean and variance: q = variance/mean, p = 1 − q, n = mean/p.
- n = 10, p = 0.3
Answer: Mean 3, variance 2.1
- Mean 4, variance 3
Answer: n = 16
Related terms
- Bernoulli trial: one trial with two outcomes
- Mode: value with highest probability
Exam tip: Check: variance must be less than the mean; if not, it isn't binomial.
Poisson distribution
The Poisson distribution models rare events in a large number of trials: P(x) = e⁻ᵐ mˣ / x!, with mean = variance = m.
What it means
- Limiting case of the binomial: n large, p small, np = m finite.
- Mean = variance = m.
- P(0) = e⁻ᵐ; P(1) = m e⁻ᵐ.
- Examples: printing errors per page, accidents per day, calls per minute.
- m = 2
Answer: e⁻²
- n = 1,000, p = 0.002
Answer: Poisson with m = 2
Related terms
- Recurrence: P(x + 1) = m/(x + 1) × P(x)
- Skewness: Poisson is positively skewed
Exam tip: Equal mean and variance is the giveaway for Poisson.
Normal distribution
The normal distribution is a symmetric bell curve with mean = median = mode; areas within 1, 2 and 3 SDs are about 68%, 95% and 99.7%.
What it means
- z = (x − μ)/σ converts to the standard normal (mean 0, SD 1).
- Points of inflection at μ ± σ.
- QD ≈ (2/3)σ, MD ≈ (4/5)σ.
- Total area under the curve = 1.
- x = 70, μ = 60, σ = 5
Answer: z = 2
- Within μ ± σ
Answer: 68.27%
Related terms
- Asymptotic: the curve never touches the x-axis
- Central limit theorem: sample means tend to be normal
Exam tip: Turn every normal question into z-scores first.
Memory tricks
Binomial np > npq; Poisson mean = variance = m.
Normal: 68.27%, 95.45%, 99.73% within 1, 2, 3 SDs.
QD ≈ 2/3 σ, MD ≈ 4/5 σ.
Try 3 CA Foundation-style questions
Q1. In a binomial distribution,
- the mean equals the variance
- the mean is less than the variance
- the mean is greater than the variance
- the variance is always 1
Show answer
Answer: (3) the mean is greater than the variance
Variance = npq = mean × q, and q < 1, so the variance is always less than the mean.
Q2. The standard normal distribution has
- mean 1 and standard deviation 0
- mean 100 and standard deviation 15
- mean equal to variance
- mean 0 and standard deviation 1
Show answer
Answer: (4) mean 0 and standard deviation 1
Converting with z = (x − μ)/σ gives a distribution with mean 0 and SD 1.
Q3. Marks are normally distributed with mean 60 and SD 5. The z-score of a student who scored 70 is
- 2
- 10
- −2
- 0.5
Show answer
Answer: (1) 2
z = (x − μ)/σ = (70 − 60)/5 = 2.
Practise all 12 questions Free · explanations for every option · see where you go wrong