Probability and Distribution Notes

03 May, 2026

Exam Study Guide: Probability & Distributions

Course: IIM Lucknow — IPMX Batch 2026-27 — QAB-I

Topics covered: Probability, Random Variables, Binomial, Poisson, Uniform, Exponential, Normal

Also see the solved questions from the lectures here

BLOCK 1: Probability Foundations

Core Ideas in 5 Bullets

1. Sample space (S) = list of all possible outcomes. Event (A) = a subset of S.
2. P(A) = favorable outcomes / total outcomes, with 0 ≤ P(A) ≤ 1 and P(S) = 1.
3. Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If mutually exclusive, the last term is 0.
4. Conditional probability: P(A|B) = P(A ∩ B) / P(B) — probability of A given B already happened.
5. Multiplication rule: P(A ∩ B) = P(A|B) · P(B). If independent: P(A ∩ B) = P(A) · P(B).

Bayes' Theorem

The only formula you need for inverse problems:

P(Eᵢ | D) = P(D | Eᵢ) · P(Eᵢ) / Σⱼ P(D | Eⱼ) · P(Eⱼ)

Plain English: Given you saw D, what's the chance it came from cause Eᵢ?

💡 Pattern recognition: Whenever a question gives you P(D | cause₁), P(D | cause₂)... and asks "given D, what's P(cause)?", it's Bayes. Always.

BLOCK 2: Random Variables + Uniform

Random Variable (RV) — Concept

A random variable X assigns a number to each outcome of a random experiment. Two types:

• Discrete: X takes countable values (0, 1, 2...). E.g., number of heads in 3 tosses.
• Continuous: X takes any value in an interval. E.g., heights, times, weights.

Expectation and Variance

For a discrete RV:

• E[X] = Σ x · p(x)
• V[X] = E[X²] − (E[X])², where E[X²] = Σ x² · p(x)

Linear Transformation Rule — Memorize This

If Y = aX + b (a, b are constants):

• E[Y] = a · E[X] + b
• V[Y] = a² · V[X]

+b shifts the mean but does NOT affect variance. The coefficient gets squared in variance.

Uniform Distribution — X ~ U(a, b)

PDF is flat (rectangle) on [a, b]: f(x) = 1/(b−a)

BLOCK 3: Binomial + Poisson

Binomial Distribution — X ~ B(n, p)

Use when:

• Fixed number n of independent trials
• Each trial: success (prob p) or failure (prob 1−p)
• X = number of successes

PMF: P(X = x) = C(n,x) · pˣ · (1−p)ⁿ⁻ˣ

Excel:

• =BINOM.DIST(x, n, p, FALSE) → P(X = x)
• =BINOM.DIST(x, n, p, TRUE) → P(X ≤ x)

💡 "At least one" trick: P(X ≥ 1) = 1 − P(X = 0). Most common Binomial pattern in exams.

What do x, n, and p mean?

• n = total number of trials (e.g., 5 HR directors surveyed)
• p = probability of success in one trial (e.g., 0.30 probability of saying "NOT important")
• x = number of successes (e.g., exactly 2 directors say "NOT important")

"Success" = whatever the question is counting. It does NOT always mean something good.

Poisson Distribution — X ~ Poisson(λ)

Use when:

• Counting rare events over a fixed interval (time, area, volume)
• You know the average rate λ for that interval

PMF: P(X = x) = e⁻λ · λˣ / x!

Critical scaling rule: If λ is "events per hour" and you're asked about T minutes, scale λ proportionally to match the time window.

Excel:

• =POISSON.DIST(x, λ, FALSE) → P(X = x)
• =POISSON.DIST(x, λ, TRUE) → P(X ≤ x)

How to find λ:

λ = rate × time

Example: 15 calls/hour, time = 20 minutes = 1/3 hour → λ = 15 × 1/3 = 5

BLOCK 4: Exponential + Normal

Exponential Distribution — X ~ Exp(λ)

Use when measuring waiting time (continuous time between Poisson events).

💡 Poisson-Exponential link: If events arrive Poisson with rate λ, time between events is Exponential with the same λ.

Key relationship: λ = 1 / mean

• If average time between events = 20 minutes → λ = 1/20 = 0.05

Excel: =EXPON.DIST(x, λ, TRUE) → P(X ≤ x)

Complement: P(X > x) = 1 − P(X ≤ x) — use when question asks "more than" or "after" x time.

Normal Distribution — X ~ N(μ, σ)

Continuous, symmetric. Mean = median = mode = μ. Spread controlled by σ.

Empirical rule:

• ~68% of data within μ ± 1σ
• ~95% within μ ± 2σ
• ~99.7% within μ ± 3σ

Z-score: Z = (X − μ) / σ

Converts X to Z ~ N(0,1). Z = "how many SDs above/below the mean."

Two problem types:

• Forward: Given X, find probability → compute z, look up Φ(z). Excel: =NORM.DIST(x, μ, σ, TRUE)
• Inverse: Given probability, find X → find z, solve z = (X − μ)/σ. Excel: =NORM.S.INV(p)

Critical z-values to memorize:

BLOCK 5: Final Cheat Sheet

Distribution Decision Tree

Is the variable a COUNT?

• YES → Discrete
  • Fixed n trials, each yes/no? → Binomial
  • Counting rare events in an interval? → Poisson
• NO → Continuous
  • Equally likely on [a,b]? → Uniform
  • Time between rare events? → Exponential
  • Bell-shaped, real-world measurement? → Normal

Formula Card

Linear transform Y = aX + b (any distribution):

• E[Y] = a · E[X] + b
• V[Y] = a² · V[X]

Excel Functions

The "Cumulative" Toggle

• TRUE → P(X ≤ x) — use for "at most", "less than"
• FALSE → P(X = x) — use for "exactly"

The "At Least" Trick

P(X ≥ x) = 1 − P(X ≤ x−1)

Excel (Binomial, at least 1): =1 - BINOM.DIST(0, n, p, TRUE)

When to Use "S" (Standard) Versions

• NORM.S.DIST / NORM.S.INV → when working with Z-scores (mean=0, SD=1)
• NORM.DIST / NORM.INV → when working with actual values

Five Common Mistakes to Avoid

1. Forgetting to scale Poisson λ when the time window changes — always match rate to interval first.
2. Using "at least one" the long way — go straight to 1 − P(X=0).
3. Mixing up "less than" vs "more than" in Normal — always sketch the curve and shade the region.
4. Not squaring the coefficient in variance — V[3X−4] = 9·V[X], not 3·V[X].
5. Choosing the wrong distribution — "time between events" → Exponential (not Normal). "Counts" → Poisson (not Binomial, unless n is fixed).

Before the Exam

• Excel is allowed — use the functions above. Don't compute by hand if you don't have to.
• Always sketch the distribution and shade the region you want.
• For Bayes — draw the partition + evidence diagram before writing any formula.
• For Poisson — rescale λ first, then plug in.
• For "at least one" Binomial — immediately go to 1 − (1−p)ⁿ.