CA Foundation · Quantitative Aptitude

Theoretical Distributions: formula sheet

Full chapter guide

Key formulas

Conditions for a discrete distribution: p(x) ≥ 0 for all x, and Σ p(x) = 1; Use this to find an unknown constant in a given table.
Conditions for a continuous distribution: f(x) ≥ 0, and total area under f(x) = 1; For a continuous variable, P(X = a) = 0, so P(a ≤ X ≤ b) is the same whether endpoints are included or not.
Expectation (discrete): E(X) = Σ x·p(x); Multiply each value by its probability and add.
Expectation of X²: E(X²) = Σ x²·p(x); Square the value x, not the probability.
Variance: Var(X) = E(X²) − [E(X)]²; Always non-negative. If you get a negative value, recheck your arithmetic.
Standard deviation: SD(X) = √Var(X); Same units as X.
Linear change: E(aX + b) = a·E(X) + b; Var(aX + b) = a²·Var(X); Adding a constant b does not change variance.
Cumulative probability: P(X ≤ x) = Σ p(t) for all t ≤ x; For discrete X, P(X > x) = 1 − P(X ≤ x).
Binomial probability: P(X = r) = nCr × p^r × q^(n−r), r = 0, 1, 2, ..., n; n = number of trials, r = number of successes, p = probability of success, q = 1 − p.
Combination: nCr = n! ÷ [r! × (n − r)!]; Counts the ways to place r successes among n trials. nCr = nC(n−r).
Probability of failure: q = 1 − p; p + q = 1 in every trial.
Total probability: Σ P(X = r) for r = 0 to n equals 1; Use it to check answers or to find 'at least' by subtraction.
At least one success: P(X ≥ 1) = 1 − q^n; Faster than adding P(1) + P(2) + ... + P(n).
Mean and variance: Mean = np; Variance = npq; Standard deviation = √(npq). Since q < 1 for p > 0, variance npq < mean np. They are equal only in the trivial case p = 0.
Recurrence relation: P(r + 1) = [(n − r) ÷ (r + 1)] × (p ÷ q) × P(r); Useful when you need several consecutive terms starting from P(0) = q^n.
Mean: μ = np; Average number of successes in n trials.
Variance: σ² = npq, where q = 1 − p; For 0 < p < 1, q < 1, so the variance is less than the mean.
Standard deviation: σ = √(npq); Take the square root of the variance.
Probability of r successes: P(X = r) = nCr × p^r × q^(n − r), r = 0, 1, ..., n; Used for fitting and for mode checks.
Mode: Compute (n + 1)p. If not an integer, mode = integer part. If an integer, two modes: (n + 1)p and (n + 1)p − 1; Always check whether (n + 1)p is a whole number.
Additive property: X ~ B(n1, p), Y ~ B(n2, p), independent ⇒ X + Y ~ B(n1 + n2, p); Needs the same p and independence.
Finding n and p: q = variance ÷ mean; p = 1 − q; n = mean ÷ p; Valid only when variance < mean, which holds for 0 < p < 1.
Expected frequency in fitting: Expected frequency = N × P(X = r); N is the total frequency (number of sets of trials).
Poisson probability: P(X = r) = e^(−m) × m^r ÷ r!, for r = 0, 1, 2, ...; m is the average number of occurrences in the interval. Exam questions usually give the value of e^(−m).
Mean and variance: Mean = m; Variance = m; Standard deviation = √m; Mean equals variance. This is the property most used to identify m.
Recurrence relation: P(r + 1) = m ÷ (r + 1) × P(r); Gives each probability from the previous one without computing factorials.
Poisson as limit of binomial: m = np; Use when n is large and p is small. The binomial probabilities are then close to the Poisson ones.
Probability of at least one: P(X ≥ 1) = 1 − e^(−m); Use the complement for 'at least' questions.
Sum of independent Poisson variables: If X ~ Poisson(m₁) and Y ~ Poisson(m₂) are independent, X + Y ~ Poisson(m₁ + m₂); Useful when the interval is extended or two sources are combined.
Mode: Mode = integer part of m. If m is a whole number, there are two modes: m − 1 and m; Check this when the question asks for the most likely value.
Normal variable notation: X ~ N(μ, σ²); μ is the mean, σ is the standard deviation, σ² is the variance.
Probability density function: f(x) = [1 ÷ (σ√(2π))] × e^(−(x − μ)² ÷ (2σ²)), for −∞ < x < +∞; Rarely used for calculation. You may need to recognise it. The π here is 3.14159..., and e is about 2.718.
Central equality: Mean = Median = Mode = μ; True because the curve is symmetric and has a single peak.
Skewness and kurtosis: Skewness (β₁ and γ₁) = 0; β₂ = 3; β₂ = 3 means the curve is mesokurtic.
Empirical rule: P(μ − σ < X < μ + σ) ≈ 68.27%; P(μ − 2σ < X < μ + 2σ) ≈ 95.45%; P(μ − 3σ < X < μ + 3σ) ≈ 99.73%; Exact values are 68.27%, 95.45% and 99.73%. The rounded values 68%, 95% and 99.7% are usually used in the options.
Area on each side: P(X < μ) = P(X > μ) = 0.5; Total area under the curve = 1.
Quartiles: Q₁ = μ − 0.6745σ; Q₃ = μ + 0.6745σ; Quartile deviation = 0.6745σ, which is about (2/3)σ. Mean deviation is about 0.7979σ, which is about (4/5)σ.
Quartile deviation, mean deviation and SD (approximate): QD : MD : SD ≈ 10 : 12 : 15, from QD ≈ (2/3)σ and MD ≈ (4/5)σ; This ratio is only an approximation. The exact ratio 0.6745 : 0.7979 : 1 is roughly 10.1 : 12 : 15. With σ = 15, QD ≈ 10 and MD ≈ 12.
Points of inflection: x = μ − σ and x = μ + σ; The curve changes from bending downward to bending upward at these points.
Standardisation: Z = (X − μ) ÷ σ; Z has mean 0 and SD 1. Table use is covered in the Z-table topic.
Standard normal variate: Z = (X − μ) ÷ σ; μ is the mean and σ is the standard deviation of X. Z has mean 0 and SD 1.
Symmetry: P(Z < 0) = P(Z > 0) = 0.5 and P(Z > a) = P(Z < −a); Total area under the curve is 1. Use this to handle negative Z.
Area from the mean (0-to-Z table): P(0 < Z < a) = P(−a < Z < 0) = table value at a; The table value for a negative Z is read at |Z|.
Right tail: P(Z > a) = 0.5 − P(0 < Z < a), for a ≥ 0; For a < 0, P(Z > a) = 0.5 + P(0 < Z < |a|).
Interval on opposite sides of the mean: P(−a < Z < b) = P(0 < Z < a) + P(0 < Z < b), for a, b > 0; Add the two areas.
Interval on the same side of the mean: P(a < Z < b) = P(0 < Z < b) − P(0 < Z < a), for 0 ≤ a < b; Subtract the smaller area from the larger.
Number of items: Expected number = N × probability; N is the total number of items in the group.
Standard reference areas: P(−1 < Z < 1) ≈ 0.6826, P(−2 < Z < 2) ≈ 0.9545, P(−3 < Z < 3) ≈ 0.9973; Also P(−1.96 < Z < 1.96) ≈ 0.95. Use these to check or skip table lookups.
Binomial probability: P(X = r) = nCr × p^r × q^(n−r), where q = 1 − p, r = 0, 1, ..., n; Use when n is fixed and each trial is independent with the same p.
Binomial mean and variance: Mean = np; Variance = npq; SD = √(npq); Variance is always less than the mean because q < 1.
Poisson probability: P(X = r) = e^(−m) × m^r ÷ r!, where r = 0, 1, 2, ...; m is the average number of occurrences. The values of r have no upper limit.
Poisson mean and variance: Mean = m; Variance = m; SD = √m; Mean equals variance. This is the key clue for Poisson.
Poisson as limit of binomial: m = np (use when n is large and p is small); A common rule of thumb is np less than about 5. It is an approximation, not an exact rule.
Normal distribution parameters: X ~ N(μ, σ²); mean = median = mode = μ; skewness = 0; The curve is a symmetric bell. The total area under it is 1.
Standard normal variate: Z = (X − μ) ÷ σ; Z has mean 0 and SD 1. Use it with the Z-table.
Area rule for normal curve: μ ± 1σ ≈ 68.27%; μ ± 2σ ≈ 95.45%; μ ± 3σ ≈ 99.73%; These are the standard approximate areas under the curve.
Normal approximation to binomial: X ≈ N(np, npq), so Z = (X − np) ÷ √(npq); Use when n is large and p is not near 0 or 1. A common check is that both np and nq are at least 5 or so. Apply the ±0.5 continuity correction if asked.

Quick revision

A discrete variable takes countable values; a continuous variable takes any value in a range.
The probabilities of all values of a random variable add up to 1.
Binomial needs fixed n, two outcomes, constant p and independent trials.
Binomial: P(X = r) = nCr × p^r × q^(n − r), with q = 1 − p.
Binomial mean = np and variance = npq; variance is always less than the mean.
Poisson: P(X = r) = e^(−m) × m^r ÷ r!, used for rare events.
Poisson mean and variance are both equal to m.
Normal curve is bell-shaped and symmetric about the mean; mean, median and mode are equal.
Total area under the Normal curve is 1, and half lies on each side of the mean.
Standard normal variate: Z = (X − μ) ÷ σ, with mean 0 and variance 1.
For a Normal distribution, about 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.
Binomial variance is less than its mean, Poisson has them equal, and Normal has no link between them.

Common mistakes

Forgetting to check that Σ p(x) = 1 before using a table. Fix: Add the probabilities first. If one is unknown, solve for it. This often finds the answer on its own.
Computing E(X²) as [E(X)]². Fix: Square each x first, multiply by p(x), then add. Only then subtract [E(X)]² to get variance.
Forgetting the nCr term and writing only p^r × q^(n−r). Fix: Always ask 'in how many positions can the successes occur?' and multiply by nCr.
Using p for the wrong event, such as taking p as the defective rate when the question counts good items. Fix: Success is whatever the question counts. Write 'success = ...' in your rough work before choosing p.
Using variance = np instead of npq. Fix: Remember that variance has the extra q. For 0 < p < 1, q < 1, so variance is smaller than mean.
Treating the given standard deviation as the variance. Fix: Square the SD first. Then use npq = SD².
Forgetting to change m when the interval changes. Fix: Scale m with the interval. A rate of 2 per page becomes m = 6 for 3 pages. Write 'm = ...' before using the formula.
Using m = n × q or m = p instead of m = np. Fix: For a binomial approximation, always take m = np. Here p is the probability of the rare event.
Saying the mean, median and mode of a normal distribution are different. Fix: For a normal curve, symmetry and a single peak make all three equal to μ.
Treating 68%, 95% and 99.7% as areas from the mean to μ + σ, μ + 2σ, μ + 3σ. Fix: The rule is for μ ± kσ. The one-sided area from μ to μ + σ is about 34%.

Exam tips

Questions often give a table with one unknown. Solve Σ p(x) = 1 first, because later parts depend on it.
Learn Var(aX + b) = a²Var(X) cold. It gives a quick answer without any table work.
Check whether the question asks for variance or standard deviation before you pick an option.
Expect conceptual MCQs: which variable is discrete, why P(X = a) = 0 for continuous X, or what a theoretical distribution is. Revise the definitions.
Wrong answers cost 0.25 marks. Skip a long table question if you are unsure and come back at the end.
Questions are usually direct: given n and p, find P(exactly r) or an 'at least' probability. Practise these until the setup takes seconds.
Check for the phrase 'independent' or 'with replacement'. It confirms the binomial model is meant.
Expect the same data to be used in the mean-and-variance questions, so remember np and npq with this topic.