Chapter Overview
Theoretical Distributions in One Minute
A theoretical distribution is a probability model that describes what is expected to happen. The whole chapter becomes simple once you answer one question: Is the variable counted or measured?
Factory Floor
100 bulbs are inspected. How many defectives?
→ Binomial
Telephone Exchange
Calls arrive randomly. Average 3 per minute.
→ Poisson
Student Marks
Thousands of scores cluster around the mean.
→ Normal
Random SituationWhat is observed?
→
Count or Measure?Discrete or continuous?
→
Select the ModelBinomial · Poisson · Normal
1. Probability Distribution Basics
Before learning individual distributions, understand what a probability distribution actually describes.
Core Idea
A random variable converts a random outcome into a number. A probability distribution then assigns a probability to each possible value of that random variable.
Random ExperimentTossing coins, inspecting products or measuring heights
→
Random Variable XA number representing the outcome
→
Probability DistributionProbability attached to every possible value of X
Discrete and Continuous Random Variables
Discrete Random Variable
Obtained by counting. Possible values are separate and usually whole numbers.
- Number of defectives
- Number of heads
- Number of calls
- Number of accidents
Continuous Random Variable
Obtained by measurement. It may take any value within an interval.
- Height
- Weight
- Diameter
- Time
PMF and PDF
Discrete
Probability Mass Function
Gives the probability of each exact value of a discrete random variable.
P(X = x) ≥ 0 and ΣP(X = x) = 1
Continuous
Probability Density Function
Describes how probability is distributed over intervals of a continuous variable.
f(x) ≥ 0 and total area under the curve = 1
Important Difference
For a discrete variable, P(X = x) may be positive. For a continuous variable, the probability of one exact value is zero; probability is found over an interval.
Expected Value and Variance
| Measure | Discrete Random Variable | Meaning |
|---|---|---|
| Mean or Expected Value | E(X) = ΣxP(X = x) | Long-run average value of X. |
| Variance | Var(X) = E(X²) − [E(X)]² | Average squared spread around the mean. |
| Standard Deviation | σ = √Var(X) | Spread expressed in the original unit. |
2. Distribution Selector
The first exam decision is not “Which formula?” It is “Which distribution?”
The Distribution Family
One map to remember the whole chapter
PROBABILITY MODELS
DISCRETE (Count)
Binomial
Fixed trials · n, p
Poisson
Random rate · λ
CONTINUOUS (Measure)
Normal
Bell curve · μ, σ
How to Choose a Distribution
START
Is the variable counted or measured?
COUNTED
Are trials fixed in advance?
YES
BINOMIAL
NO (rate given)
POISSON
MEASURED
Continuous values around a mean?
NORMAL
Language Clues in the Question
| Words in the Question | Likely Distribution | Why? |
|---|---|---|
| Out of 20, 50 or 100 items | Binomial | Fixed number of trials. |
| Pass/Fail, Good/Defective, Head/Tail | Binomial | Two possible outcomes. |
| Per minute, per day, per page | Poisson | Random occurrences per interval. |
| Average number of accidents/calls/errors | Poisson | Rate λ is given. |
| Height, weight, marks, diameter | Normal | Continuous measurement. |
Exam Trap
The word “defective” does not decide the model. “Defectives in a fixed sample of 100” is Binomial, while “defects per metre of fabric” is generally Poisson.
3. Binomial Distribution
Binomial Distribution counts the number of successes in a fixed number of identical independent trials.
Exam-Ready Definition
A Binomial Distribution is a discrete probability distribution that gives the probability of obtaining exactly x successes in n independent Bernoulli trials when the probability of success p remains constant.
A Binomial Distribution is a discrete probability distribution that gives the probability of obtaining exactly x successes in n independent Bernoulli trials when the probability of success p remains constant.
Bernoulli Trial — Storyboard
Imagine a machine producing bulbs. Each bulb is inspected once.
One Trial
Inspect one bulb
Success or Failure
Success or Failure
Probabilities
P(Defective) = p
P(Good) = q = 1−p
P(Good) = q = 1−p
Repeat n Times
n independent,
identical inspections
identical inspections
Count Successes
X = number of
defectives (0 to n)
defectives (0 to n)
Five Conditions
Fixed Trials
n is decided in advance.
Two Outcomes
Success or failure.
Independent
One trial does not affect another.
Constant p
Same probability every trial.
Count Successes
X records selected outcomes.
Success Need Not Be Desirable
If defective items are being counted, “defective” is treated as success for that question.
Building the Binomial Formula
Choose locations
nCx
×
x successes
px
×
n−x failures
qn−x
=
Probability
P(X=x)
P(X = x) = nCx px qn−x, where x = 0, 1, 2, …, n
| Symbol | Meaning |
|---|---|
| n | Total number of trials. |
| x | Required number of successes. |
| p | Probability of success in one trial. |
| q | Probability of failure, q = 1 − p. |
| nCx | Number of arrangements of x successes among n trials. |
Language of Probability
Exactly xP(X = x)
At Most xP(X ≤ x)
At Least xP(X ≥ x)
More Than xP(X > x)
Fast Translation
“At least 2” means 2 or more. “At most 2” means 0, 1 or 2. “More than 2” begins from 3.
Solved Illustration 1 — Guided Ladder
Question
A sample contains 5 bulbs. Probability that a bulb is defective is 0.1. Find P(exactly 2 defectives).
Given
n = 5, x = 2, p = 0.1 → q = 0.9
Think
Fixed trials + two outcomes + constant p → Binomial
Formula
P(X = 2) = 5C2 (0.1)2 (0.9)3
Calculate
10 × 0.01 × 0.729 = 0.0729
Answer
P(X = 2) = 0.0729
Exam Tip
Always write n, x, p, q first. It prevents exponent mistakes.
Solved Illustration 2 — Complement
Question
A fair coin is tossed 4 times. Find the probability of at least one head.
Think
Direct addition is lengthy. Use complement: 1 − P(X = 0).
Calculate
P(X = 0) = (1/2)4 = 1/16 → 1 − 1/16 = 15/16
Answer
15/16
4. Binomial Mean, Variance, Shape and Properties
Meanμ = np
Varianceσ² = npq
Standard Deviationσ = √npq
Quick Relationship
Since q ≤ 1, npq ≤ np. Therefore, the variance of a Binomial Distribution is never greater than its mean.
Maximum Variance
For a fixed n, variance npq is maximum when p = q = 0.5.
Small Illustration
If n=20 and p=0.5Variance = 20×0.5×0.5 = 5
If n=20 and p=0.2Variance = 20×0.2×0.8 = 3.2
ConclusionVariance is larger at p=0.5.
Shape of Binomial Distribution
Symmetrical
When p = q = 0.5, the distribution is symmetrical.
Positively Skewed
When p < q, the longer tail lies to the right.
Negatively Skewed
When p > q, the longer tail lies to the left.
Mode
Mode is the integral part of (n+1)p.
If (n+1)p is an integer, there are two modes: (n+1)p and (n+1)p − 1.
If (n+1)p is an integer, there are two modes: (n+1)p and (n+1)p − 1.
Additive Property
X₁ ~ B(n₁,p), X₂ ~ B(n₂,p) ⇒ X₁ + X₂ ~ B(n₁ + n₂, p)
Finding n and p from Mean and Variance
Question
Mean = 6, Variance = 3. Find n and p.
Given
np = 6, npq = 3
Calculate
q = 3/6 = 0.5 → p = 0.5 → n = 6/0.5 = 12
Answer
n = 12, p = 0.5
5. Fitting a Binomial Distribution
Fitting means calculating the theoretical frequencies expected under a Binomial model.
Find total frequency N.Add all observed frequencies.
Find n.n is the highest possible value of x.
Find the mean.Mean = Σfx / N.
Find p.Since mean = np, p = mean / n.
Find q.q = 1 − p.
Calculate theoretical frequencies.Expected frequency = N × P(X = x).
Mini Illustration
Suppose N = 100, n = 2 and p = 0.3. Then q = 0.7.
x = 0100 × (0.7)² = 49
x = 1100 × 2 × 0.3 × 0.7 = 42
x = 2100 × (0.3)² = 9
Check49 + 42 + 9 = 100
6. Poisson Distribution
Poisson Distribution counts random occurrences over a specified interval when an average rate is known.
Exam-Ready Definition
A Poisson Distribution is a discrete probability distribution that gives the probability of a specified number of occurrences of a random event in a fixed interval of time, space or area.
A Poisson Distribution is a discrete probability distribution that gives the probability of a specified number of occurrences of a random event in a fixed interval of time, space or area.
Where Poisson Appears in Real Life
📞 Telephone
Calls arrive randomly. Average rate λ per minute is known.
🏭 Quality
Defects per metre of fabric or per batch of components.
🚗 Traffic
Number of accidents on a stretch of road per day.
📖 Printing
Misprints per page in a book.
Poisson Probability Function
P(X = x) = e−λ λx / x!, where x = 0, 1, 2, …
Meanλ
Varianceλ
Standard Deviation√λ
Signature Property
For Poisson Distribution, Mean = Variance = λ.
Mode
If λ is not an integer → Mode = greatest integer less than λ.
If λ is an integer → two modes: λ and λ − 1.
If λ is an integer → two modes: λ and λ − 1.
Poisson Approximation to Binomial
When n is large, p is small and np remains finite, Poisson approximates Binomial with λ = np.
Question
A telephone exchange receives an average of 3 calls per minute. Find P(exactly 2 calls).
Given
λ = 3, x = 2
Formula
P(X = 2) = e−3 × 3² / 2!
Calculate
9e−3 / 2 ≈ 9 × 0.0498 / 2 ≈ 0.2241
Answer
≈ 0.2241
Discrete Variable Trap
If a Poisson variable is written as X < 2.7, possible integer values are 0, 1 and 2.
7. Fitting a Poisson Distribution
Find total frequency N.Add observed frequencies.
Find mean λ.λ = Σfx / N.
Find the first probability.P(0) = e−λ.
Use recurrence where convenient.P(x+1) = P(x) × λ / (x+1).
Find expected frequencies.N × P(x).
Mini Illustration
If N = 200 and λ = 1:
P(0)e−1 ≈ 0.3679 → Expected ≈ 74
P(1)P(0) × 1/1 ≈ 0.3679 → Expected ≈ 74
P(2)P(1) × 1/2 ≈ 0.1839 → Expected ≈ 37
8. Normal Distribution
Normal Distribution is the most important continuous distribution. It is used for measurements that cluster around an average.
68.28%Within μ ± σ
95.46%Within μ ± 2σ
99.73%Within μ ± 3σ
Normal PDF
f(x) = 1/(σ√2π) × e−(x−μ)²/(2σ²)
f(x) = 1/(σ√2π) × e−(x−μ)²/(2σ²)
Parameters
Mean μ
Determines the centre (location) of the curve.
Standard Deviation σ
Determines the spread (width) of the curve.
Characteristics of the Normal Curve
Bell-Shaped
Rises to one central peak.
Symmetrical
Both sides are mirror images.
Mean = Median = Mode
All three at the centre.
Total Area = 1
Entire curve = total probability.
Half Area Each Side
0.5 left and 0.5 right of μ.
Asymptotic Tails
Approaches but never touches axis.
Points of Inflexion
Located at μ − σ and μ + σ.
Continuous
Probability over intervals only.
9. Standard Normal Distribution and Z-Scores
Z = (X − μ) / σ
Interpretation
Z = 2 means the value lies 2 standard deviations above the mean. Z = −1.5 means it lies 1.5 standard deviations below the mean.
How to Solve Area Questions
Draw a rough bell curve.Mark the mean and the required values.
Convert X-values to Z-values.Use Z = (X − μ) / σ.
Identify the required region.Left tail, right tail, central area or between two values.
Use the Normal table.Read the relevant area carefully.
Apply symmetry or subtraction.Total area is 1; each half is 0.5.
Question
Marks ~ N(60, 10). Find Z for 75 marks.
Formula
Z = (X − μ) / σ = (75 − 60) / 10
Answer
Z = 1.5
Important Standard Normal Values
| Z-Value | Common Use |
|---|---|
| 1.645 | Approximately 5% in one tail. |
| 1.96 | Approximately 2.5% in each tail; central 95%. |
| 2.33 | Approximately 1% in one tail. |
| 2.58 | Approximately 0.5% in each tail; central 99%. |
10. Binomial, Poisson and Normal — Comparison
| Point | Binomial | Poisson | Normal |
|---|---|---|---|
| Nature | Discrete | Discrete | Continuous |
| Main Use | Successes in fixed trials | Random events per interval | Measurements around a mean |
| Parameters | n and p | λ | μ and σ |
| Mean | np | λ | μ |
| Variance | npq | λ | σ² |
| Examples | Defectives in 100 items | Calls per minute | Height or diameter |
ICAI Exam Traps — Poster
Binomial Traps
- Success need not be favourable.
- q = 1 − p (always).
- Exponent of q is n − x.
- “At least” ≠ “at most”.
- Variance never exceeds mean.
Poisson & Normal Traps
- Poisson: Mean = Variance = λ.
- Decimal limits → convert to integers.
- Normal: P(X = exact value) = 0.
- Each side of μ has area 0.5.
- Standard Normal: mean 0, S.D. 1.
Revision Mind Map
Theoretical Distributions — Branching Recall
Identify the variable first. Fixed trials → Binomial · Random rate → Poisson · Continuous measure → Normal.
THEORETICAL DISTRIBUTIONS
BINOMIAL
Fixed n · Two outcomes · Constant p
P(X=x) = nCx pˣ qⁿ⁻ˣ
Mean = np · Variance = npq
Mode ≈ (n+1)p · Max Var at p=0.5
Fitting · Parameter finding · Complement
POISSON
Random events per interval
P(X=x) = e⁻λ λˣ / x!
Mean = Variance = λ
Approx. Binomial when n large, p small
Fitting via recurrence relation
NORMAL
Continuous · Bell-shaped · Symmetrical
Mean = Median = Mode = μ
Z = (X − μ) / σ
68% · 95% · 99.7% rules
Table areas · Symmetry · Tails
Final Memory: Fixed trials → Binomial | Random occurrences → Poisson | Continuous measurement → Normal | Always convert Normal values to Z before using the table.