ISS PREVIOUS YEAR 2016 PAPER-1 SET-A The lifetime of a bulb follows an exponential distribution with mean 100 hours. The bulb is switched on for exactly 4 hours every day and remains switched off for the remaining time. What is the probability that the bulb stops working on or before the 25th day? Options: (a) (1 − e⁻¹) / (1 − e⁻¹⁄²⁵) (b) 1 − e⁻¹⁄²⁵ (c) 1 − e⁻¹ (d) e⁻¹

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A To compare the lifetimes of bulbs produced by two companies A and B, one bulb from each company was selected at random and their lifetimes were observed. Assume that the lifetimes follow exponential distributions with: Mean(A) = 1000 daysMean(B) = 800 days What is the probability that the bulb from company B fails before the bulb from company A? Options: (a) 4/9 (b) 5/9 (c) 25/81 (d) 25/41

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A 10 balls are placed in 10 boxes independently and uniformly at random. Initially, all boxes are empty. What is the expected number of boxes that remain empty?

A simple random sample of size 10 is selected with replacement from a population of size 100. What is the expected number of unique elements in the sample? Options:(a) 100 × (99/100)¹⁰(b) 100 × [1 − (99/100)¹⁰](c) 100 × (9/10)¹⁰(d) 100 × (1 − 99/100)¹⁰

Let XXX be a random variable with probability generating functionP(s)=∑kpkskP(s) = \sum_{k} p_k s^kP(s)=∑k​pk​sk. Find the probability generating function of Y=2XY = 2XY=2X.

Question Let XXX be a Poisson variate with parameter λ\lambdaλ such that P(X = 2) = 2P(X = 4) + 20P(X = 6) What is the coefficient of skewness? Options: (a) 1/√3(b) 1(c) 1/2(d) −1/√3

Question:Which of the following statements are associated with residuals of a regression model? The sum of the residuals in any regression model that contains an intercept is always zero. The sum of the observed values yiy_iyi​ is not equal to the sum of the fitted values y^i\hat{y}_iy^​i​. The sum of the residuals weighted by the corresponding regressor variable is zero. The sum of the residuals weighted by the corresponding fitted values is zero. Select the c

Let the random variable X have the distribution: P(X = 0) = P(X = 3) = p P(X = 1) = 1 − 3p P(X = 2) = p where 0 ≤ p ≤ 1/2 Find the maximum value of V(X).

A Poisson random variable X has mean equal to 1/2. Let Y = 2X. Consider the following: E(Y) = 1 Var(Y) = 4 μ3(Y) = 4 μ4(Y) = 28 Which of the above is/are correct?

For the distribution: f(x) = [1 / B(p, q)] × [x^(p − 1) / (1 + x)^(p + q)],where 0 < x < ∞, p > 0, q > 0, find the harmonic mean. Options: (a) p / (p + q) (b) 1 / p (c) (p − 1) / q (d) (p + 1) / (q − 1)

X₁ and X₂ are independent Poisson random variables such that: P(X₁ = 2) = P(X₁ = 1)P(X₂ = 2) = P(X₂ = 3) Find the variance of (X₁ − 2X₂).

If in 6 trials, X is a binomial variate which follows the relation 9 P(X = 4) = P(X = 2) then what is the probability of success? Options:(a) 1/8 (b) 1/4 (c) 3/8 (d) 3

Let X have a Bernoulli distribution with mean 0.4. What is the variance of (2X − 3)? Options: (a) 0.24(b) 0.48(c) 0.60(d) 0.96

ISS PREVIOUS YEAR PAPER-1 2016 SET A Q.NO.-1 Let (X, Y) be jointly distributed with density f(x, y) = e^(-y), for 0 < x < y < ∞ OR f(x, y) = 0, otherwise Consider the following statements: E(X) = 1 E(Y) = 2 E(XY) = 2 Which of the above are correct? (a) 1 and 2 only(b) 2 and 3 only(c) 1 and 3 only(d) 1, 2 and 3