Maharashtra Forest Services 2012 Statistics (Optional) Exam Questions

Q1. 1(a) Let X be a gamma random variable (r.v.) with shape parameter α and scale 10 and parameter λ. Obtain the moment generating function (mgf) of X.

Q2.

(a) Answer the following sub-questions : Let \X_n, n ≥ 1\ be a sequence of i.i.d. r.v.s. with the common m.g.f. M(t). Let N be a Poisson r.v. with mean λ. Let S_N = X₁ + \ldots + X_N and S_o = 0. Obtain the m.g.f. of S_N, when N is independent of \X_n\. Suppose X is a Poisson r.v. with mean λ and Y is Binomial (m, λ/m), and X and Y

(b) are independent. Derive a formula for P[X = Y].

(c) Let X be an exponential r.v. such that P[X ≤ 2] = 0.2. Obtain the mean of X.
(i) Obtain P[X > 2 + S/X > S]. Justify your answer.
(ii)
3. Answer the following sub-questions : Let the joint density of (X, Y) be (a) f(x, y) = 2/(1 + x + y) 3 , x>0, y>0 = 0 otherwise Obtain the marginal density of X.
(i) Determine whether X and Y are independent.
(ii) Let X₁, X₂, \ldots, X₁₀₀ be independent Poisson r.v.s. with mean
4. Using the (b) appropriate approximation and tables obtain : P[X₁ + X₂ + \ldots + X₁₀₀ ≤ 450] Let X be a r.v. with p.d.f. f
(x) = (e^-xxⁿ)/(n!), x > 0, f
(x) = 0, x ≤ 0, and n a positive
(c) integer. Let Y be a Poisson r.v. with mean λ. Show that P[X ≥ λ] = P[Y ≤ n]. SECTION - B
4. Answer the following sub-questions : For a simple random sampling without replacement from a population of size N, (a) let \bary denote the sample mean. Show that N\bary is an unbiased estimator of the population total. In a RBD with 4 treatments and blocks, derive a test of hypothesis of equality of (b) any two treatment effects. Let X₁ and X₂ be two independent exponential r.v.s. with mean 1/λ and let
(c) U₁ = min (X₁, X₂), U₂ = max (X₁, X₂). Show that the r.v.s. V₁ = 2U₁ and V₂ = U₂ - U₁ are independent and obtain their distribution.

Q2. (a) Answer the following sub-questions : Let \X_n, n ≥ 1\ be a sequence of i.i.d. r.v.s. with the common m.g.f. M(t). Let N be a Poisson r.v. with mean λ. Let S_N = X₁ + \ldots + X_N and S_o = 0. Obtain the m.g.f. of S_N, when N is independent of \X_n\. Suppose X is a Poisson r.v. with mean λ and Y is Binomial (m, λ/m), and X and Y

(a) Answer the following sub-questions : Let \X_n, n ≥ 1\ be a sequence of i.i.d. r.v.s. with the common m.g.f. M(t). Let N be a Poisson r.v. with mean λ. Let S_N = X₁ + \ldots + X_N and S_o = 0. Obtain the m.g.f. of S_N, when N is independent of \X_n\. Suppose X is a Poisson r.v. with mean λ and Y is Binomial (m, λ/m), and X and Y

(b) are independent. Derive a formula for P[X = Y].

(c) Let X be an exponential r.v. such that P[X ≤ 2] = 0.2. Obtain the mean of X.
(i) Obtain P[X > 2 + S/X > S]. Justify your answer.
(ii)
3. Answer the following sub-questions : Let the joint density of (X, Y) be (a) f(x, y) = 2/(1 + x + y) 3 , x>0, y>0 = 0 otherwise Obtain the marginal density of X.
(i) Determine whether X and Y are independent.
(ii) Let X₁, X₂, \ldots, X₁₀₀ be independent Poisson r.v.s. with mean
4. Using the (b) appropriate approximation and tables obtain : P[X₁ + X₂ + \ldots + X₁₀₀ ≤ 450] Let X be a r.v. with p.d.f. f
(x) = (e^-xxⁿ)/(n!), x > 0, f
(x) = 0, x ≤ 0, and n a positive
(c) integer. Let Y be a Poisson r.v. with mean λ. Show that P[X ≥ λ] = P[Y ≤ n]. SECTION - B
4. Answer the following sub-questions : For a simple random sampling without replacement from a population of size N, (a) let \bary denote the sample mean. Show that N\bary is an unbiased estimator of the population total. In a RBD with 4 treatments and blocks, derive a test of hypothesis of equality of (b) any two treatment effects. Let X₁ and X₂ be two independent exponential r.v.s. with mean 1/λ and let
(c) U₁ = min (X₁, X₂), U₂ = max (X₁, X₂). Show that the r.v.s. V₁ = 2U₁ and V₂ = U₂ - U₁ are independent and obtain their distribution.

• (a) Answer the following sub-questions : Let \X_n, n ≥ 1\ be a sequence of i.i.d. r.v.s. with the common m.g.f. M(t). Let N be a Poisson r.v. with mean λ. Let S_N = X₁ + \ldots + X_N and S_o = 0. Obtain the m.g.f. of S_N, when N is independent of \X_n\. Suppose X is a Poisson r.v. with mean λ and Y is Binomial (m, λ/m), and X and Y
• (b) are independent. Derive a formula for P[X = Y].
• (c) Let X be an exponential r.v. such that P[X ≤ 2] = 0.2. Obtain the mean of X.

Q3. 1
(c) In a CRD with 4 treatments, each having 5 replicates, obtain an expression for the residual sum of squares (R0²) and state, giving reasons, its distribution.

Q4. 1(d) Let X1, X2, ..., Xn be a random sample from a distribution with pdf f
(x) = θ (1 - x)^{θ - 1}, 0 < x < 1, θ >
1. Find the maximum likelihood estimator (MLE) of θ.

Q5. 1(d) Minimize z = 5x1 + 2x2 subject to x1 - x2 ≥ 3; 2x1 + 3x2 ≥ 5; x1, x2 ≥ 0. State one feasible solution for each problem. Are the solutions you have stated optimal for both the problems?

Q6. 1(e) Let X1, X2, X3, X4, X5 be independent normal r.v.s with mean μ and variance σ^2. Let U = (1/9) { (2X1 - X2 - X3)^2 + (2X2 - X1 - X3)^2 + (2X3 - X1 - X2)^2 }. State, giving reasons, the distribution of W = U / (X4 - X5)^2. Obtain 'a' such that P[W > a] = 0.05.

Q7. 2(a) Let {X_n, n ≥ 1} be a sequence of i.i.d. r.v.s. with the common mgf M(t). Let N be a Poisson r.v. with mean λ. Let S_N = X₁ + ... + X_N and S₀ = 0. Obtain the mgf of S_N, when N is independent of {X_n}.

Q8. 2(b) Suppose X is a Poisson r.v. with mean λ and Y is Binomial(m, λ/m), and X and Y are independent. Derive a formula for P[X = Y].

Q9. 3(a)
(i) Let the joint density of (X, Y) be f(x, y) = 2/(1 + x + y)^3, x > 0, y > 0; = 0 otherwise. Obtain the marginal density of X.

Q10. 3(a)
(ii) Determine whether X and Y are independent.

Q11. 3(b) Let X1, X2, ..., X100 be independent Poisson r.v.s with mean 4. Using the appropriate approximation and tables obtain: P[X1 + X2 + ... + X100 ≤ 450].