ISS 2016 Statistics Paper-1 Solution: Question 34 (Chi-Square to Beta Distribution)
Continuing our question-by-question walk through the ISS Statistics Paper-1 previous year papers, today we pick up right where the last post left off and move to Question 34 of the 2016 paper. This one looks intimidating at first glance because it bundles three separate distributional claims into a single question, but once you see the underlying pattern, all three fall into place together.
Quick Summary
Topic:Relationship between independent Chi-square variates and the Beta Type I and Beta Type II (Beta Prime) distributions
Question Reference:ISS 2016, Statistics Paper-1, Question 34
Correct Answer:(d) 1, 2 and 3 — all three statements are correct
The Question, As Asked
34. Consider the following statements:
X1 and X2 are independent χ2 variates with n1 and n2 d.f. respectively, then Y = X1/X2 follows β2(n1/2, n2/2) distribution.
X1 and X2 are independent χ2 variates with n1 and n2 d.f. respectively, then Z = X1/(X1 + X2) follows β1(n1/2, n2/2) distribution.
X1 and X2 are independent standard normal distribution, then W = X12 + X22 follows χ2(2) distribution.
Which of the above statements is/are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 3 only
(d) 1, 2 and 3
Step 1: Writing the Chi-Square Density as a Gamma Density
Every one of these three statements is really the same trick applied three different ways, so it pays to set up the right tool once. If a random variable U follows χ2(n), then U is exactly a Gamma random variable with shape n/2 and scale 2. Its density is:
f(u) = [1 / (2n/2 Γ(n/2))] · u(n/2) − 1 · e−u/2, for u > 0
So X1 ~ χ2(n1) has density f1(x1) built the same way with n1, and X2 ~ χ2(n2) has density f2(x2) built with n2. Since X1 and X2 are independent, their joint density is just f1(x1) · f2(x2). Everything in statements 1 and 2 follows from transforming this joint density.
Step 2: Checking Statement 1 — Y = X1/X2
To find the distribution of Y = X1/X2, we cannot work with Y alone — we need a second, auxiliary variable so that the transformation from (X1, X2) to (Y, Z) is one-to-one and invertible. Take Z = X2 as the auxiliary variable. Then:
X1 = YZ, X2 = Z
The Jacobian of this transformation works out to |∂(x1, x2)/∂(y, z)| = z. Substituting X1 = yz and X2 = z into the joint density and multiplying by the Jacobian z gives the joint density of (Y, Z):
f(y, z) = C · y(n1/2) − 1 · z((n1+n2)/2) − 1 · e−z(y+1)/2
where C = 1 / [2(n1+n2)/2 Γ(n1/2) Γ(n2/2)]. Now Y's marginal density is obtained by integrating z out from 0 to ∞. The integral is a standard Gamma-function integral:
∫0∞ z((n1+n2)/2) − 1 e−z(y+1)/2 dz = Γ((n1+n2)/2) · [2/(1+y)](n1+n2)/2
Multiplying this back into f(y,z) and simplifying the powers of 2, all the constants collapse neatly into a Beta function:
f(y) = [1 / B(n1/2, n2/2)] · y(n1/2) − 1 · (1+y)−(n1/2 + n2/2), y > 0
This is precisely the density of a Beta Type II (Beta Prime) distribution with parameters (n1/2, n2/2) — written β2(n1/2, n2/2). So Statement 1 is correct.
Step 3: Checking Statement 2 — Z = X1/(X1 + X2)
The setup is almost identical, but this time we use W = X1 + X2 as the auxiliary variable, so that:
X1 = ZW, X2 = (1 − Z)W
Here the Jacobian is |∂(x1, x2)/∂(z, w)| = w. Substituting into the joint density of X1 and X2 and multiplying by w gives:
f(z, w) = C · z(n1/2) − 1 (1−z)(n2/2) − 1 · w((n1+n2)/2) − 1 e−w/2
Integrating w out from 0 to ∞ produces the same Gamma-function form as before, but this time it does not depend on z at all, so it comes out of the integral as a pure constant. What remains for z is:
f(z) = [1 / B(n1/2, n2/2)] · z(n1/2) − 1 (1−z)(n2/2) − 1, 0 < z < 1
This is the classical Beta Type I density with parameters (n1/2, n2/2), written β1(n1/2, n2/2). So Statement 2 is correct.
Step 4: Checking Statement 3 — W = X12 + X22
This one needs far less machinery. If X is a standard normal variable, then X2 follows χ2(1) — this is the defining construction of the chi-square distribution with one degree of freedom. Since X1 and X2 are independent standard normals, X12 and X22 are two independent χ2(1) variates.
The additive property of the chi-square distribution says that if two independent chi-square variates are added, their degrees of freedom simply add up: χ2(n1) + χ2(n2) = χ2(n1+n2). Here n1 = n2 = 1, so:
W = X12 + X22 ~ χ2(1+1) = χ2(2)
So Statement 3 is correct as well.
Pro Tip: The trickiest part of this question isn't any single derivation — it's remembering which ratio gives which Beta type. A simple memory anchor: if the denominator of the ratio is the
Final Answer
All three statements — the Beta Type II result for X1/X2, the Beta Type I result for X1/(X1+X2), and the additive chi-square result for X12+X22 — are correct. This means the correct option is:
(d) 1, 2 and 3
This matches the official ISS 2016 answer key, which also lists option (d) as correct for Question 34.
Why This Question Matters
This question is really testing whether you understand the family relationships between the Gamma, Chi-square, Beta Type I, Beta Type II, and eventually the F-distribution. In fact, if you divide Statement 1's ratio by n1/n2, you get exactly the F-distribution with (n1, n2) degrees of freedom — so the derivation you just worked through is also secretly the derivation of where the F-distribution comes from. This is one of the most frequently recurring derivation patterns across ISS, ISI, and other statistics services exams, so a solid, from-scratch understanding of it pays off well beyond this one question.
Frequently Asked Questions
What is the actual difference between Beta Type I and Beta Type II distributions?
Beta Type I (the “ordinary” Beta distribution) is defined on the interval (0, 1) and typically arises from a ratio of the form X/(X+Y). Beta Type II, also called the Beta Prime distribution, is defined on (0, ∞) and arises from a plain ratio X/Y of two independent Gamma or chi-square variates. Both share the same two shape parameters and the same Beta function normalizing constant, but their support and interpretation are different.
How is this question connected to the F-distribution?
The F-distribution with (n1, n2) degrees of freedom is defined as F = (X1/n1)/(X2/n2) for independent χ2(n1) and χ2(n2) variates. Statement 1's variable Y = X1/X2 is just F multiplied by the constant n1/n2, so the Beta Type II density we derived here is a simple reparametrization away from the standard F density used in ANOVA and variance-ratio tests.
Why does the square of a standard normal variable follow chi-square with 1 degree of freedom?
This is essentially the definition of the chi-square distribution: if Z ~ N(0,1), then Z2 is defined to follow χ2(1). More generally, a chi-square distribution with n degrees of freedom is built as the sum of n independent squared standard normal variables, which is exactly why adding two independent χ2(1) variables gives χ2(2).
Does the ratio result in Statement 1 only work for chi-square variates, or is it more general?
It is more general. The derivation only used the fact that chi-square variates are a special case of the Gamma distribution. The same result holds for any two independent Gamma random variables sharing a common scale parameter — their ratio always follows a Beta Type II distribution with shape parameters equal to the two Gamma shape parameters, and the ratio of one to their sum always follows Beta Type I.
What is the most common mistake students make with this type of question?
The most common mistake is mixing up which ratio produces which Beta type, or forgetting that Beta Type II lives on (0, ∞) rather than (0, 1) and then trying to force it into the standard Beta Type I formula. Another frequent slip is dividing the numerator and denominator by n1 and n2 out of habit (as in the F-distribution) when the question, as stated here, does not ask for that normalization at all.
Where does this topic typically appear in the ISS Statistics syllabus?
This falls under the theory of sampling distributions, specifically the derived distributions of functions of Gamma and chi-square variates, which is a core part of the Probability and Distribution Theory portion of ISS Statistics Paper-1. It also underlies later syllabus topics like the F-test, ANOVA, and variance-ratio testing, so examiners return to this family of derivations often.
If any step above felt unclear, or if you'd solve this one differently, drop a comment below — and if you know a fellow ISS aspirant working through the same 2016 paper, consider sharing this post with them so you can compare notes.
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