Quick Summary: The official GATE DA syllabus names nine probability distributions — four discrete (uniform, Bernoulli, binomial, Poisson) and five continuous (uniform, exponential, normal, standard normal, t, chi-squared). Every distribution question in the paper reduces to three skills: recognise which distribution the story describes, recall its mean and variance, and compute one probability. This cheat sheet gives you all three — the master table, the recognition rules, the special properties GATE loves, and two fully worked NAT problems.
By Piyush Wairale — Instructor, BS Data Science program at IIT Madras · IIT Madras alumnus · 10,000+ GATE students mentored · Last updated: August 2026
Key Takeaways
- Binomial(n, p): mean np, variance np(1−p) — “number of successes in n independent trials”. Poisson(λ): mean = variance = λ — “count of rare events per interval”.
- The exponential distribution is the only continuous memoryless distribution: P(X > s+t | X > s) = P(X > t) — “waiting time until the next event”.
- Any normal N(μ, σ²) converts to the standard normal via Z = (X − μ)/σ; the 68–95–99.7 rule gives the areas within 1, 2 and 3 standard deviations.
- Binomial(n, p) → Poisson(np) when n is large and p small; the t-distribution has heavier tails than the normal and approaches it as degrees of freedom grow.
- Chi-squared with k degrees of freedom is the distribution of a sum of k squared standard normals — mean k, variance 2k — which is why it appears in variance tests.
On this page
The master table · Recognition rules · Special properties GATE tests · Worked NAT 1: binomial · Worked NAT 2: normal standardization · Common traps · GATE patterns · FAQs
Distribution questions are the most formula-driven marks in GATE DA: if you can name the distribution and recall two numbers, the question is already half solved. This cheat sheet is the drilling companion to our Probability & Statistics pillar guide — everything here goes one level deeper on the distributions themselves, with the properties and worked NATs that a survey post cannot fit. For lectures and tests on the full subject, see the Probability & Statistics for GATE DA Course.
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The Master Table (Your One-Page Revision Sheet)
| Distribution | PMF / PDF | Mean | Variance | Models |
|---|---|---|---|---|
| Discrete uniform {1,…,n} | P(X=k) = 1/n | (n+1)/2 | (n²−1)/12 | Fair die, random pick |
| Bernoulli(p) | px(1−p)1−x, x∈{0,1} | p | p(1−p) | One yes/no trial |
| Binomial(n, p) | C(n,k)pk(1−p)n−k | np | np(1−p) | Successes in n trials |
| Poisson(λ) | e−λλk/k! | λ | λ | Rare-event counts per interval |
| Continuous uniform [a,b] | f(x) = 1/(b−a) | (a+b)/2 | (b−a)²/12 | Equally likely over a range |
| Exponential(λ) | λe−λx, x ≥ 0 | 1/λ | 1/λ² | Waiting time to next event |
| Normal N(μ, σ²) | bell curve about μ | μ | σ² | Sums/averages of many effects |
| Standard normal N(0,1) | φ(z), tabulated | 0 | 1 | The Z-table reference |
| t (ν d.o.f.) | heavy-tailed bell | 0 (ν>1) | ν/(ν−2), ν>2 | Means with unknown σ, small n |
| Chi-squared (k d.o.f.) | Σ of k squared Z’s | k | 2k | Variance tests, goodness of fit |
Recognition Rules: Reading the Story
- “Number of successes/defects/heads in n independent attempts” → Binomial. One attempt only → Bernoulli.
- “Number of arrivals/calls/errors per minute/page/km” → Poisson (a rate, an interval, no fixed n).
- “Time until the next arrival/failure” → Exponential (the waiting-time twin of the Poisson count).
- “Chosen uniformly at random between a and b” → Continuous uniform; from a finite equally-likely set → discrete uniform.
- “Heights/measurement errors/averages of many samples” → Normal (the CLT signature).
- “Sample mean with σ estimated from a small sample” → t. “Sample variance, or comparing observed vs expected counts” → chi-squared.
Practise this mapping on twenty one-line stories until it is instant — GATE frequently awards a full mark purely for the recognition step.
The Special Properties GATE Actually Tests
Exponential memorylessness (with the one-line proof)
P(X > s+t | X > s) = P(X > s+t)/P(X > s) = e−λ(s+t)/e−λs = e−λt = P(X > t). A component that has already survived s hours behaves like a brand-new one — no ageing. The exponential is the only continuous distribution with this property (the geometric is its discrete counterpart). GATE phrases it as “a bulb has lasted 100 hours; what is the probability it lasts 50 more?” — answer: exactly P(X > 50), ignore the history.
Poisson: mean = variance, and the binomial limit
Both the mean and variance of Poisson(λ) equal λ — so a question stating “a count has mean 4 and variance 4” is announcing a Poisson. And when n is large, p small, with np moderate, Binomial(n, p) ≈ Poisson(np): 1,000 items each defective with probability 0.002 → defect count ≈ Poisson(2). GATE uses this to make otherwise-ugly binomial computations easy — recognising the approximation is the question.
Normal standardization and the 68–95–99.7 rule
Every normal question becomes a Z-table question through Z = (X − μ)/σ. Anchors to memorise: P(Z ≤ 1.96) ≈ 0.975, P(Z ≤ 1.645) ≈ 0.95, P(Z ≤ 1) ≈ 0.8413; and the empirical rule — about 68% of mass within μ±σ, 95% within μ±2σ, 99.7% within μ±3σ. Symmetry does the rest: P(Z ≤ −a) = 1 − P(Z ≤ a).
t vs normal, and chi-squared as squared Z’s
The t-distribution is symmetric like the normal but with heavier tails — more probability of extreme values — converging to N(0,1) as degrees of freedom grow (practically indistinguishable past ν ≈ 30). The chi-squared with k degrees of freedom is by definition Z₁² + … + Zk² for independent standard normals — hence right-skewed, positive-only, mean k, variance 2k. These two facts explain why the t-test and chi-squared test in the hypothesis-testing section use those reference distributions.
Worked NAT 1: Binomial
Question. A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads (round to 4 decimals)?
Solution. X ~ Binomial(5, 0.5). P(X = 3) = C(5,3)·(0.5)³·(0.5)² = 10 × 0.125 × 0.25 = 10/32 = 0.3125. Extension GATE adds: “at least 4 heads” = P(4) + P(5) = 5/32 + 1/32 = 6/32 = 0.1875 — always convert “at least/at most” into a short sum, and use the complement when the sum is long (P(X ≥ 1) = 1 − P(0)).
Worked NAT 2: Normal Standardization
Question. Exam scores are normally distributed with mean 60 and standard deviation 10. Given P(Z ≤ 1.5) = 0.9332, what fraction of students score between 45 and 75?
Solution. Standardise both ends: for X = 75, Z = (75−60)/10 = 1.5; for X = 45, Z = (45−60)/10 = −1.5. So P(45 ≤ X ≤ 75) = P(−1.5 ≤ Z ≤ 1.5) = P(Z ≤ 1.5) − P(Z ≤ −1.5) = 0.9332 − (1 − 0.9332) = 0.9332 − 0.0668 = 0.8664. The rhythm never changes: standardise, sketch the region, use symmetry, subtract. GATE always supplies the exact Z-table values you need — if your Z’s don’t match the given table entries, recheck the standardisation.
Common Traps
- Exponential parameterisation: mean is 1/λ, not λ. “Average waiting time 10 minutes” means λ = 0.1 per minute.
- Variance vs standard deviation: N(60, 25) usually means variance 25, σ = 5 — but read the notation the question defines.
- Continuous point probabilities: P(X = a) = 0 for continuous X, so P(X ≤ a) = P(X < a); for discrete X they differ — “at most” vs “less than” matters.
- Binomial complement laziness: computing P(X ≥ 2) term by term when 1 − P(0) − P(1) is three lines shorter.
- Poisson interval scaling: a rate of 3 per hour is 1.5 per half hour — rescale λ to the question’s interval before computing.
- Poisson’s placement in the syllabus: the official list groups it with the continuous PDFs, but Poisson is discrete — a trivia-style MCQ has hinged on this.
How GATE Tests Distributions
- Direct computation NATs (2 marks): a binomial/Poisson/normal probability with numbers chosen to be calculator-friendly — the two worked problems above are the templates.
- Recognition MCQs (1 mark): which distribution models the described scenario; or “mean equals variance” style property identification.
- Parameter recovery (1–2 marks): given a mean and variance, find n and p of a binomial (np and np(1−p) give two equations), or λ of an exponential.
- Property statements (1 mark): memorylessness, t vs normal tails, chi-squared d.o.f. arithmetic, the binomial→Poisson limit.
- CLT hybrids (2 marks): standardising a sample mean with σ/√n — the bridge into the confidence-interval material.
Master Probability the Structured Way
All nine distributions with lectures, solved GATE PYQs, topic-wise tests and doubt support:
- Probability & Statistics for GATE DA Course & Test Series — by IIT Madras alumnus Piyush Wairale.
- Complete GATE DA 2027 Course — all 7 subjects, 250+ hours, 4,000+ questions and full-length AI mock tests.
FAQs on Probability Distributions for GATE DA
Which distributions are in the GATE DA syllabus?
Discrete: uniform, Bernoulli, binomial (and Poisson, listed alongside the continuous family). Continuous: uniform, exponential, normal, standard normal, t-distribution and chi-squared — nine in total, all covered in the master table above.
Do I need to memorise the normal PDF formula?
No — GATE normal questions are solved by standardisation and supplied Z-table values, never by integrating the density. Memorise Z = (X−μ)/σ, the symmetry rules, and the 68–95–99.7 anchors instead.
What is the most-tested distribution fact?
Poisson’s mean = variance = λ, closely followed by exponential memorylessness and the binomial mean/variance pair (np, np(1−p)). All three have appeared repeatedly across GATE papers.
How do distributions connect to the rest of the paper?
The normal drives confidence intervals and the z-test; t and chi-squared drive the other two tests; Bernoulli underlies logistic regression and naive Bayes in Machine Learning; and conditional probabilities over these distributions feed Bayes’ theorem questions.
How should I revise this in the last week?
Rebuild the master table from memory once a day (it takes five minutes), then solve two recognition drills and two NATs. The table plus the recognition rules cover essentially every distribution mark in the paper.
Nine distributions, two numbers each, six recognition rules, five properties — that is the entire distribution game in GATE DA. Rebuild the table until it lives in your hands, and these become the fastest marks you earn on exam day. Continue with the Bayes’ theorem deep dive or the full Probability & Statistics guide.
Make Every Probability Mark Count
All nine distributions to hypothesis testing — lectures, solved PYQs, sectional tests and mentorship by Piyush Wairale (IIT Madras).
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