UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 Statistics-I
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary, and indicate the same clearly.
1. (a) The lifetime of a mobile charger (in hours) has the normal distribution with mean (μ) = 100 and variance (σ²) = 400.
(i) What is the probability that the mobile charger lasts at least 125 hours?
(ii) If the mobile charger has already lasted for 105 hours, what is the conditional probability that it will last another 20 hours?
(Normal Distribution Table is given in Page Nos. 10 and 11) 4+4=8
(b) (i) State Lindeberg condition for non-identically distributed independent variables to hold central limit theorem (CLT). 4
(ii) Examine whether CLT holds for the sequence {X_n}, where
P{X_n = ± (1/2^n)} = 1/2
(c) Suppose X₁, X₂, …, X_n are independently identically distributed (iid) observations from a location parameter family with cumulative distribution function F(x − θ), −∞ < θ < ∞. Show that R = X_(n) − X_(1) is ancillary statistic, where X_(n) = max_i {X_i} and X_(1) = min_i {X_i}. 8
UPSC Indian Forest Service (IFoS) MAIN Previous Year Question Paper 2024 – statistics-II
(d) Consider the problem of testing H₀: θ = 1 versus H₁: θ = 1/2, where θ is the mean of a Poisson random variable. Let X and Y be a random sample from Poisson(θ) distribution. Consider the following test procedure:
Reject H₀ if X = 1 or (Y = 1 and X + Y ≤ 2), otherwise accept H₀. Determine the probability of type I and type II errors. 8
(e) In an ecological study of the feeding behaviour of birds, the number of hops between flights is counted for several birds:
No. of hops | Observed frequency
1 | 48
2 | 31
3 | 20
4 | 9
5 | 6
6 | 5
7 | 4
8 | 2
9 | 1
10 | 1
11 | 1
12 | 1
Total | 130
Assuming that the data are generated by a geometric(p) model and take a uniform prior for p, what is the posterior distribution of parameter p? What are the mean and the standard deviation of the posterior distribution? 4+2+2=8
2. (a) (i) Suppose that the transition probability matrix of a Markov chain model is given by
[IMAGE: transition probability matrix missing/not visible]
Compute n-step transition probability matrix (P^n).
(ii) A random variable X has mean (μ) = 40 and standard deviation (σ) = 2. Apply Chebyshev’s inequality to estimate P(25 ≤ X ≤ 55).
(b) Let X₁, X₂, X₃, … be a sequence of independent random variables, where the probability mass function (PMF) of the random variables X_n, n = 1, 2, 3, … is given by
P(X_n = ± √n/2) = 1/2
Does the law of large numbers hold for this sequence of random variables? 10
(c) (i) Let X follow binomial b(4, θ) distribution, 0 < θ < 1. To test the hypothesis H₀: 1/3 ≤ θ ≤ 1/2 versus H₁: θ < 1/3 or θ > 1/2 with size 0.3, the test function is based on the following procedure:
Reject H₀ with probability γ₁, if X = 1
Reject H₀ with probability γ₂, if X = 3
Determine the constants γ₁ and γ₂. Compute power of the test when θ = 0.2. 10
(ii) Let X₁, X₂, …, X_n be independently identically distributed (iid) random variables from exponential distribution with mean θ > 0. Define
g(X) = X_n / (X₁ + X₂ + … + X_n)
Find E_θ[g(X)].
3. (a) Two random variables X and Y have the following joint probability density function:
[IMAGE: joint probability density function missing/not visible]
Find the following:
(i) Constant k
(ii) Marginal density function of X and Y
(iii) E(X|Y = 1)
(iv) V(X|Y = 1)
(v) cov(X, Y)
(b) The following data represent systolic blood pressure (SBP) of 15 individuals before and after participating in a physical exercise programme:
S. No. | Before | After
1 | 127 | 120
2 | 134 | 136
3 | 140 | 132
4 | 132 | 134
5 | 129 | 128
6 | 130 | 137
7 | 122 | 126
8 | 127 | 107
9 | 129 | 132
10 | 138 | 142
11 | 137 | 128
12 | 134 | 130
13 | 141 | 134
14 | 138 | 132
15 | 133 | 125
Use Wilcoxon signed-rank test to test the difference in SBP after participating in physical exercise as compared to before at 5% level of significance.
[Given that W_(15, 0.05) = 25, W_(14, 0.05) = 21 for two-tailed test]
(c) (i) A 24-hour advance prediction of a day’s high temperature is ‘unbiased’ if the long-term average of the error in prediction (true high temperature minus predicted high temperature) is zero. The errors in predictions (x) made by one meteorological station for 20 randomly selected days were recorded. The results were
∑_{i=1}^{20} x_i = −15, ∑_{i=1}^{20} x_i² = 35
Assume normal distribution of errors and test the null hypothesis that the predictions are unbiased versus the alternative that they are biased at 1% level of significance. Verify whether the decision would be the same at 5% and 10% level of significance.
[Given that t_(19, 0.005) = 2.861, t_(19, 0.025) = 2.093, t_(19, 0.05) = 1.729, where P(t_n > t_(n, α)) = α] 3+1+1=5
(ii) Distinguish between Wald’s SPRT and the test based on Neyman-Pearson theory to test simple null versus simple alternative hypotheses.
4. (a) Let X and Y be independent Poisson variables with V(X + Y) = 9 and
P(X = 3 | X + Y = 6) = 5/54
Obtain E(Y).
(b) Suppose X and Y are independent identically distributed as exponential variates with mean = 1. Obtain the characteristic function of X and X − Y. Hence, deduce the distribution of Z = X − Y.
(c) Suppose that X is a discrete random variable with
P(X = 0) = 2θ/3, P(X = 1) = θ/3, P(X = 2) = 2(1 − θ)/3 and P(X = 3) = (1 − θ)/3
where θ ∈ [0,1] is the parameter. The following 10 independent observations were taken from such distribution:
{3, 0, 2, 1, 3, 2, 1, 0, 2, 1}
(i) Find the moment estimate of θ.
(ii) Find the approximate standard error for your estimate.
(iii) What is the maximum likelihood estimate (MLE) of θ?
(iv) What is the approximate standard error of the MLE of θ?
## SECTION-B
5. (a) For the model Y = β₀ + β₁X + ε, ε ∼ N(0, σ²), a regression line is fitted on the basis of n paired observations (x_i, y_i), i = 1, …, n. The fitted line is y = b₀ + b₁x, where
var(b₀) = σ² ∑_{i=1}^n x_i² / ∑_{i=1}^n (x_i − x̄)², var(b₁) = σ² / ∑_{i=1}^n (x_i − x̄)² and cov(b₀, b₁) = −σ² x̄ / ∑_{i=1}^n (x_i − x̄)²
For a given x = x₀, one can predict the value of a new observation Y₀ as Ŷ₀ = b₀ + b₁x₀. Find an expression for var(Ŷ₀ − Y₀) and compare it with var(Ŷ₀).
Find c_n, the standard deviation of (Ŷ₀ − Y₀)/σ.
(b) Consider a p-dimensional random vector X with mean μ and variance-covariance matrix Σ. Let (u₁, …, u_p) be an orthonormal system of eigenvectors of Σ with corresponding eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_p. Denote U = (u₁, …, u_p). Let Y = (Y₁, …, Y_p)′ be the vector of principal components of X. Then for any pair (i, j) ∈ {1, …, p}, show that
cov(X_i, Y_j) = u_{ij} λ_j
ρ(X_i, Y_j) = u_{ij} √λ_j / √var(X_i)
where u_{ij} = (u_j)_i = (i, j)th element of U.
(c) From the following population based on SRSWOR scheme with n = 3, verify that sample mean ȳ and sample variance s² are unbiased for population mean Ȳ and variance S². Also show that the sampling variance of ȳ agrees with the expression for V(ȳ) = (N − n)/N · S²/n:
i | Y_i
1 | 5
2 | 8
3 | 3
4 | 11
5 | 9
(d) Define main effects and interaction effects in a 2² factorial experiment. Consider an experiment with two factors, reactant concentration (A) and catalyst (B). Let the two levels of factor A be 15 percent (a₀) and 25 percent (a₁) concentration. The two levels of factor B are 2 pounds (b₁) and 1 pound (b₀) of the catalyst. The order in which the runs are made is random. The data obtained are as follows:
Treatment combination | Replication I | II | III | Total
a₀b₀ | 28 | 25 | 27 | 80
a₁b₀ | 36 | 32 | 32 | 100
a₀b₁ | 18 | 19 | 23 | 60
a₁b₁ | 31 | 30 | 29 | 90
Obtain the best estimates of main effects and interaction effects. Test for the significance of main effects and interaction effects. Set up the ANOVA table.
(e) Observations Y₁, …, Y_n are described by the model Y_i = β x_i² + ε_i, where x₁, …, x_n are fixed constants and ε₁, …, ε_n are iid Normal(0, σ²). Find the following:
(i) Least squares estimate of β
(ii) Maximum likelihood estimate of β
(iii) Best unbiased estimate of β
6. (a) The sample mean vector and covariance matrix, computed on the basis of a random sample of size 20 from a bivariate normal population N₂(μ, Σ), are given below:
[IMAGE: sample mean vector and covariance matrix missing/not visible]
(i) Evaluate the test statistic for testing H₀: μ = (μ₁, μ₂)′ = (7, 11)′.
(ii) Test H₀ at 5% level of significance. What conclusion can be reached?
(iii) Find the simultaneous confidence interval for μ₁.
(b) What is orthogonal factor model in factor analysis? Write down an orthogonal factor model with m common factors together with the covariance structure.
Consider three standardized random variables Z₁, Z₂ and Z₃ with a single factor (m = 1):
[IMAGE: factor model equations missing/not visible]
where var(F₁) = 1, cov(ε, F₁) = 0 and
[IMAGE: covariance structure missing/not visible]
Obtain the covariance matrix.
(c) Suppose an analyst uses the model y_i = β₀* + β₁*x_i + ε_i* instead of the true model y_i = β₀ + β₁x_i + β₂x_i² + β₃x_i³ + ε_i with cov(y) = σ²I, y = (y₁, …, y₇)′.
(i) Obtain E(β̂₀*) and E(β̂₁*) (in terms of true model parameters), if the observations are taken at x = −3, −2, −1, 0, 1, 2, 3. Note here that β̂₀* and β̂₁* are the least square estimates of β₀* and β₁*.
(ii) Find E(s₁²) for the same set of values of x, where s₁² is the sample variance for the true model.
7. (a) (i) For the following design, write the C-matrix (information matrix of the design) and obtain independent estimable treatment contrasts:
[IMAGE: design layout missing/not visible]
The number denotes the treatments. Also examine whether it is a connected design. 10
(ii) Define confounding. Suppose that 2⁴ = 16 treatments cannot be run using one batch of raw materials. The experimenter can run eight treatment combinations from a single batch of materials. A 2⁴ confounded in two blocks is run. Construct a design with two blocks of eight observations each with ABCD confounded. 5
(b) (i) Describe the probability proportional to size sampling with replacement scheme. Obtain an unbiased estimator for the population total along with its sampling variance under this scheme. 10
(ii) Estimate the gain in efficiency due to PPS sampling compared to SRS sampling based on SRS sample. 5
(c) In sampling with unequal probabilities, without replacement a sample of size 2 is drawn. The first unit is drawn with PPS and the second unit with PPS of remaining units. Show that Yates, Grundy and Sen’s variance estimator is always positive for this sampling system. 10
8. (a) (i) Let a′α is a vector of (ν − t) independent estimable treatment contrasts from an incomplete block design. Obtain the test statistic for testing the hypothesis H₀: a′α = 0. 7
(ii) Obtain the best estimate of treatment effect and derive the test statistic for testing the linear hypothesis of equality of treatment effects given yield from a balanced incomplete block design. 8
(b) Describe two-stage sampling. Obtain the estimator of population total when SRSWOR is used at both the stages. 10
(c) Show that under the usual assumptions for one-way ANOVA model y_{ij} = μ_i + ε_{ij}, j = 1, …, n_i, i = 1, …, k with E(ε_{ij}) = 0, var(ε_{ij}) = σ², cov{ε_{ij}, ε_{i′j′}} = 0, i ≠ i′, j ≠ j′, ε_{ij}’s are iid N(0, σ²) ∀ i, j
P[∑_{i=1}^k c_i ȳ_i − A₀ √(S_p² ∑_{i=1}^k c_i²/n_i) ≤ ∑_{i=1}^k c_i μ_i ≤ ∑_{i=1}^k c_i ȳ_i + A₀ √(S_p² ∑_{i=1}^k c_i²/n_i)] = 1 − α
simultaneously for all c = (c₁, …, c_k), where
A₀ = √(K F / k ∑_{i=1}^k (n_i − 1)), α
S_p² = 1/k ∑_{i=1}^k ∑_{j=1}^{n_i} (y_{ij} − ȳ_i)²
Normal Distribution Table
z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09
0.00 | 0.5000 | 0.5040 | 0.5080 | 0.5120 | 0.5160 | 0.5199 | 0.5239 | 0.5279 | 0.5319 | 0.5359
0.10 | 0.5398 | 0.5438 | 0.5478 | 0.5517 | 0.5557 | 0.5596 | 0.5636 | 0.5675 | 0.5714 | 0.5753
0.20 | 0.5793 | 0.5832 | 0.5871 | 0.5910 | 0.5948 | 0.5987 | 0.6026 | 0.6064 | 0.6103 | 0.6141
0.30 | 0.6179 | 0.6217 | 0.6255 | 0.6293 | 0.6331 | 0.6368 | 0.6406 | 0.6443 | 0.6480 | 0.6517
0.40 | 0.6554 | 0.6591 | 0.6628 | 0.6664 | 0.6700 | 0.6736 | 0.6772 | 0.6808 | 0.6844 | 0.6879
0.50 | 0.6915 | 0.6950 | 0.6985 | 0.7019 | 0.7054 | 0.7088 | 0.7123 | 0.7157 | 0.7190 | 0.7224
0.60 | 0.7257 | 0.7291 | 0.7324 | 0.7357 | 0.7389 | 0.7422 | 0.7454 | 0.7486 | 0.7517 | 0.7549
0.70 | 0.7580 | 0.7611 | 0.7642 | 0.7673 | 0.7704 | 0.7734 | 0.7764 | 0.7794 | 0.7823 | 0.7852
0.80 | 0.7881 | 0.7910 | 0.7939 | 0.7967 | 0.7995 | 0.8023 | 0.8051 | 0.8078 | 0.8106 | 0.8133
0.90 | 0.8159 | 0.8186 | 0.8212 | 0.8238 | 0.8264 | 0.8289 | 0.8315 | 0.8340 | 0.8365 | 0.8389
1.00 | 0.8413 | 0.8438 | 0.8461 | 0.8485 | 0.8508 | 0.8531 | 0.8554 | 0.8577 | 0.8599 | 0.8621
1.10 | 0.8643 | 0.8665 | 0.8686 | 0.8708 | 0.8729 | 0.8749 | 0.8770 | 0.8790 | 0.8810 | 0.8830
1.20 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 | 0.8944 | 0.8962 | 0.8980 | 0.8997 | 0.9015
1.30 | 0.9032 | 0.9049 | 0.9066 | 0.9082 | 0.9099 | 0.9115 | 0.9131 | 0.9147 | 0.9162 | 0.9177
1.40 | 0.9192 | 0.9207 | 0.9222 | 0.9236 | 0.9251 | 0.9265 | 0.9279 | 0.9292 | 0.9306 | 0.9319
1.50 | 0.9332 | 0.9345 | 0.9357 | 0.9370 | 0.9382 | 0.9394 | 0.9406 | 0.9418 | 0.9429 | 0.9441
1.60 | 0.9452 | 0.9463 | 0.9474 | 0.9484 | 0.9495 | 0.9505 | 0.9515 | 0.9525 | 0.9535 | 0.9545
1.70 | 0.9554 | 0.9564 | 0.9573 | 0.9582 | 0.9591 | 0.9599 | 0.9608 | 0.9616 | 0.9625 | 0.9633
1.80 | 0.9641 | 0.9649 | 0.9656 | 0.9664 | 0.9671 | 0.9678 | 0.9686 | 0.9693 | 0.9699 | 0.9706
1.90 | 0.9713 | 0.9719 | 0.9726 | 0.9732 | 0.9738 | 0.9744 | 0.9750 | 0.9756 | 0.9761 | 0.9767
2.00 | 0.9772 | 0.9778 | 0.9783 | 0.9788 | 0.9793 | 0.9798 | 0.9803 | 0.9808 | 0.9812 | 0.9817
2.10 | 0.9821 | 0.9826 | 0.9830 | 0.9834 | 0.9838 | 0.9842 | 0.9846 | 0.9850 | 0.9854 | 0.9857
2.20 | 0.9861 | 0.9864 | 0.9868 | 0.9871 | 0.9875 | 0.9878 | 0.9881 | 0.9884 | 0.9887 | 0.9890
2.30 | 0.9893 | 0.9896 | 0.9898 | 0.9901 | 0.9904 | 0.9906 | 0.9909 | 0.9911 | 0.9913 | 0.9916
2.40 | 0.9918 | 0.9920 | 0.9922 | 0.9925 | 0.9927 | 0.9929 | 0.9931 | 0.9932 | 0.9934 | 0.9936
2.50 | 0.9938 | 0.9940 | 0.9941 | 0.9943 | 0.9945 | 0.9946 | 0.9948 | 0.9949 | 0.9951 | 0.9952
2.60 | 0.9953 | 0.9955 | 0.9956 | 0.9957 | 0.9959 | 0.9960 | 0.9961 | 0.9962 | 0.9963 | 0.9964
2.70 | 0.9965 | 0.9966 | 0.9967 | 0.9968 | 0.9969 | 0.9970 | 0.9971 | 0.9972 | 0.9973 | 0.9974
2.80 | 0.9974 | 0.9975 | 0.9976 | 0.9977 | 0.9977 | 0.9978 | 0.9979 | 0.9979 | 0.9980 | 0.9981
2.90 | 0.9981 | 0.9982 | 0.9982 | 0.9983 | 0.9984 | 0.9984 | 0.9985 | 0.9985 | 0.9986 | 0.9986
3.00 | 0.9987 | 0.9987 | 0.9987 | 0.9988 | 0.9988 | 0.9989 | 0.9989 | 0.9989 | 0.9990 | 0.9990
3.10 | 0.9990 | 0.9991 | 0.9991 | 0.9991 | 0.9992 | 0.9992 | 0.9992 | 0.9992 | 0.9993 | 0.9993
3.20 | 0.9993 | 0.9993 | 0.9994 | 0.9994 | 0.9994 | 0.9994 | 0.9994 | 0.9995 | 0.9995 | 0.9995
3.30 | 0.9995 | 0.9995 | 0.9995 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997
3.40 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9998
3.50 | 0.9998 | 0.9998 | 0.9998 | 0.9998 | 0.9998 | 0.9998 | 0.9998 | 0.9998 | 0.9998 | 0.9998
3.60 | 0.9998 | 0.9998 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999
3.70 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999
3.80 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999
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