43  Probability distributions: Binomial, poisson and normal distributions

43.1 Random Variables and Distributions

A random variable is a real-valued function defined on the sample space of a random experiment. Its probability distribution describes how probabilities are assigned to its possible values. Random variables come in two flavours: discrete (countable values — number of heads, defectives, customer arrivals) and continuous (uncountable values — height, weight, time, price). This topic covers the three most important distributions in business statistics: the Binomial and Poisson (discrete) and the Normal (continuous). They are the workhorses of probability and the gateway to inferential statistics.

43.2 Binomial Distribution

The Binomial distribution describes the number of successes in n independent Bernoulli trials, each with probability of success p.

TipBinomial — Conditions
  • Fixed number of trials n.
  • Each trial has only two outcomes — success or failure.
  • Constant probability p of success.
  • Trials are independent.

43.2.1 Probability Mass Function (PMF)

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n\]

43.2.2 Mean and Variance

TipBinomial Parameters
Parameter Value
Mean np
Variance npq = np(1−p)
Standard Deviation √(npq)
Mode ≈ np
Range 0 to n

Note: For binomial, mean > variance since q < 1.

43.2.3 Examples

  • Coin tosses — number of heads in n tosses (p = 0.5).
  • Quality control — number of defectives in a batch of n.
  • Marketing — number of customers responding positively out of n contacted.

43.3 Poisson Distribution

The Poisson distribution describes the number of events occurring in a fixed interval of time or space, when events occur independently at a constant average rate λ. Discovered by Siméon Denis Poisson (1837).

43.3.1 Probability Mass Function

\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k = 0, 1, 2, \ldots\]

43.3.2 Mean and Variance

TipPoisson Parameters
Parameter Value
Mean λ
Variance λ
Standard Deviation √λ
Range 0 to ∞

Mean = Variance is the defining property of the Poisson distribution.

43.3.3 Conditions for Poisson

TipPoisson — Conditions
  • Discrete events in a continuous interval.
  • Independent — one event doesn’t affect another.
  • Constant rate λ in non-overlapping intervals.
  • Rare events — many trials, small p, finite λ = np.

43.3.4 Binomial → Poisson Approximation

When n is large (n → ∞) and p is small (p → 0) with np = λ fixed, the Binomial distribution approaches the Poisson distribution with mean λ.

43.3.5 Examples

  • Number of customer arrivals per hour at a bank counter.
  • Number of telephone calls in a switchboard per minute.
  • Defects in a long bolt of cloth.
  • Goals scored in a football match.
  • Accidents at an intersection per day.

43.4 Normal (Gaussian) Distribution

The Normal distribution is the most important continuous distribution. Discovered by Abraham de Moivre (1733), developed by Carl Friedrich Gauss (1809), applied to errors by Laplace. Its bell-shape arises from the Central Limit Theorem — the sum/mean of many independent random variables tends to normal regardless of their original distributions.

43.4.1 Probability Density Function

\[f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-(x - \mu)^2 / 2\sigma^2}, \quad -\infty < x < \infty\]

43.4.2 Properties

TipProperties of Normal Distribution
  • Bell-shaped, symmetric about μ.
  • Mean = Median = Mode = μ.
  • Skewness = 0; Kurtosis β₂ = 3 (mesokurtic).
  • Defined by two parameters: μ (mean) and σ (SD).
  • Total area under the curve = 1.
  • Asymptotic to x-axis — never touches.
  • Empirical Rule (68-95-99.7):
    • ≈ 68 % within μ ± 1σ
    • ≈ 95 % within μ ± 2σ
    • ≈ 99.7 % within μ ± 3σ.

43.4.3 Standard Normal Distribution

The standard normal has μ = 0 and σ = 1, denoted Z. Any normal variable is converted:

\[Z = \frac{X - \mu}{\sigma}\]

Z-tables give cumulative probabilities of the standard normal.

43.4.4 Central Limit Theorem (CLT)

Central Limit Theorem: For a random sample of size n from a population with mean μ and finite variance σ², the sampling distribution of the sample mean tends to be normal with mean μ and SD σ/√n as n → ∞, regardless of the population’s distribution.

Rule of thumb: n ≥ 30 usually suffices.

TipThree Distributions at a Glance
Property Binomial Poisson Normal
Type Discrete Discrete Continuous
Parameters n, p λ μ, σ
Range 0 to n 0 to ∞ −∞ to +∞
Mean np λ μ
Variance npq λ σ²
Symmetric? Only if p = 0.5 No (skewed for small λ) Always
Limiting form → Poisson (np = λ, n→∞, p→0) → Normal (large λ) — 

flowchart LR
  B[Binomial<br/>n, p] -->|n→∞, p→0, np=λ| P[Poisson λ]
  P -->|λ → ∞| N[Normal μ, σ]
  B -->|n→∞, p=0.5| N
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

NoteDistractor warning

PYQ trap: Poisson — Mean = Variance = λ. Binomial — Mean (np) > Variance (npq) because q < 1. Normal — Mean, Median, Mode coincide.

43.5 Other Distributions — Brief

TipOther Common Distributions
  • Bernoulli — single trial with two outcomes (special case of Binomial with n = 1).
  • Geometric — number of trials until first success.
  • Negative Binomial — number of trials until r-th success.
  • Hypergeometric — sampling without replacement.
  • Uniform (Rectangular) — equal probability over an interval.
  • Exponential — time between Poisson events; memoryless.
  • t-distribution — Student’s t (Gosset 1908); used when σ is unknown.
  • Chi-square (χ²) — sum of squared standard normals; goodness-of-fit, independence.
  • F-distribution — ratio of two chi-squares; ANOVA.

43.6 Practice Questions

Q 01BinomialEasy

Mean of a Binomial distribution is:

  • Anp
  • Bnpq
  • C√(np)
  • Dλ
View solution
Correct Option: A
**Mean = np**; variance = npq.
Q 02PoissonEasy

Mean and variance of Poisson distribution are:

  • ABoth equal to np
  • BBoth equal to λ
  • CEqual to μ and σ²
  • D0 and 1
View solution
Correct Option: B
**Mean = Variance = λ** — Poisson's defining property.
Q 03Standard NormalEasy

The standard normal distribution has:

  • Aμ = 1, σ = 0
  • Bμ = 0, σ = 1
  • Cμ = 1, σ = 1
  • Dμ = 0, σ = 0
View solution
Correct Option: B
Z ~ N(0, 1).
Q 0468-95-99.7Medium

Approximately 95 % of observations of a normal distribution lie within:

  • A±1σ of mean
  • B±2σ of mean
  • C±3σ of mean
  • D±4σ of mean
View solution
Correct Option: B
Empirical Rule: **95 % within ±2σ**.
Q 05CLTMedium

The Central Limit Theorem says the sampling distribution of the sample mean is:

  • AAlways Binomial
  • BApproximately Normal for large n, regardless of population distribution
  • CAlways Poisson
  • DAlways identical to population
View solution
Correct Option: B
CLT — sum/mean ⇒ approx normal for large n; mean μ, SD σ/√n.
Q 06B→PHard

Binomial approaches Poisson when:

  • An is large and p is large
  • Bn is large and p is small with np = λ
  • Cn is small and p is large
  • DBoth n and p approach 1
View solution
Correct Option: B
Limiting form: rare events, **large n, small p, np = λ**.
Q 07ComputeMedium

For Binomial with n = 10, p = 0.5: mean and SD are:

  • A5 and 1.58
  • B5 and 2.5
  • C10 and 1.58
  • D2.5 and 5
View solution
Correct Option: A
np = 5; SD = √(npq) = √2.5 ≈ **1.58**.
Q 08Normal AuthorMedium

The normal distribution is associated with:

  • APoisson
  • BCarl Friedrich Gauss (and de Moivre, Laplace)
  • CBernoulli
  • DPearson
View solution
Correct Option: B
**Gauss** popularised; de Moivre (1733) first derived; Laplace developed.
Q 09Z-scoreMedium

For X ~ N(50, 10²), the Z-score of X = 70 is:

  • A0.2
  • B2.0
  • C7.0
  • D20
View solution
Correct Option: B
Z = (70 − 50)/10 = **2.0**.
Q 10Poisson useMedium

Which is **best modeled** by Poisson?

  • ANumber of heads in 10 coin tosses
  • BNumber of customer arrivals per hour at a bank
  • CHeight of college students
  • DTime between arrivals
View solution
Correct Option: B
Counts of independent events per fixed interval — Poisson.
Q 11SymmetryEasy

Normal distribution is:

  • ARight-skewed
  • BLeft-skewed
  • CSymmetric / Bell-shaped
  • DUniform
View solution
Correct Option: C
Bell-shaped, symmetric about μ.
Q 12ParametersMedium

Match each distribution with its parameters:

Distribution Parameters
(i) Binomial (a) λ
(ii) Poisson (b) μ, σ
(iii) Normal (c) n, p
  • A(i)-(c), (ii)-(a), (iii)-(b)
  • B(i)-(a), (ii)-(b), (iii)-(c)
  • C(i)-(b), (ii)-(c), (iii)-(a)
  • D(i)-(c), (ii)-(b), (iii)-(a)
View solution
Correct Option: A
Binomial — n,p; Poisson — λ; Normal — μ,σ.
Q 13CLT nMedium

Rule of thumb for n in the CLT:

  • An ≥ 5
  • Bn ≥ 10
  • Cn ≥ 30
  • Dn ≥ 1000
View solution
Correct Option: C
**n ≥ 30** generally adequate.
Q 14SEHard

Standard error of the sample mean is:

  • Aσ × √n
  • Bσ / √n
  • Cσ²
  • Dn / σ
View solution
Correct Option: B
**SE(x̄) = σ/√n** — by CLT.
Q 15OtherMedium

The *t-distribution* was developed by:

  • AKarl Pearson
  • BWilliam Gosset (Student) 1908
  • CR.A. Fisher
  • DGauss
View solution
Correct Option: B
Gosset under pen-name *Student*, while at Guinness (1908).
Q 16ContinuousEasy

Which is a **continuous** distribution?

  • ABinomial
  • BPoisson
  • CNormal
  • DBernoulli
View solution
Correct Option: C
Normal is continuous; others are discrete.
Q 17Mode of NormalMedium

In a normal distribution:

  • AMean ≠ Median ≠ Mode
  • BMean = Median = Mode
  • CMean > Median > Mode
  • DMode > Mean
View solution
Correct Option: B
All three coincide at μ.
Q 18Poisson AuthorMedium

Siméon Denis Poisson discovered the distribution in:

  • A1733
  • B1837
  • C1908
  • D1933
View solution
Correct Option: B
**Poisson 1837** in *Recherches sur la probabilité…*.
Q 19BernoulliEasy

A Bernoulli distribution is a special case of:

  • ABinomial with n = 1
  • BPoisson with λ = 1
  • CNormal with σ = 1
  • DUniform on [0, 1]
View solution
Correct Option: A
Single Bernoulli trial = Binomial(1, p).
Q 20ExponentialHard

Time *between* successive Poisson events follows:

  • ANormal
  • BExponential
  • CBinomial
  • DUniform
View solution
Correct Option: B
**Exponential** distribution — memoryless.

43.7 Quick Recall

ImportantQuick recall
  • Random variable: discrete (Binomial, Poisson) vs continuous (Normal).
  • Binomial(n, p): PMF \(\binom{n}{k} p^k q^{n-k}\); Mean = np; Variance = npq.
  • Poisson(λ): PMF \(e^{-\lambda} \lambda^k / k!\); Mean = Variance = λ.
  • Binomial → Poisson when n→∞, p→0, np = λ.
  • Normal(μ, σ²): bell-shaped, symmetric; Mean = Median = Mode = μ. Empirical Rule 68-95-99.7.
  • Standard normal Z = (X − μ)/σ ~ N(0,1).
  • Central Limit Theorem — sample mean → Normal(μ, σ/√n) for n ≥ 30.
  • t-distribution (Gosset 1908 “Student”); χ² (sum of squared standard normals); F (ratio of χ²s).
  • Exponential — time between Poisson events; memoryless.
  • Bernoulli — Binomial with n = 1.