Q1. Which of the following statements are True or False? Give short proof or counter example in your answer.

• (i)) If the correlation coefficient between X and Y is -0.8, then the correlation coefficient between 2X-1 and -3Y-1is -0.48. (90 words)
• (ii)) If X and Y are independent binomial variates with parameters (n, p₁) and (n₂, p₂) respectively, then X + Y has binomial distribution with parameters (n₁ + n₂, p₁ + p₂). (90 words)
• (iii)) The function defined as f(x) = [|x|; -1<x<1, [0; otherwise is a probability density function. (90 words)
• (iv)) For a normal distribution with mean μ and variance σ², the hypotheses H₁ : μ = μ₀, σ² = 1and H₂ : μ = μ₀, σ² ≥ 1 are simple hypotheses. (90 words)
• (v)) In a problem of testing of a simple hypothesis against a simple alternative, if the probability of type-I error is known to be 0.06, then the power of the test will be 0.94. (90 words)

Q2. The mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find

• (i)) the percentage of children having I.Q. under 80. (225 words)
• (ii)) the limits in which the I.Q. of the middle 40% of the children will lie. You may like to use the following values: P(Z > 1.25) = 0.1056 P(Z <-0.525) = 0.3 (225 words)

Q3. 6 observations on (X,Y) yielded the following data: ΣX = 30, ΣY = 180, ΣXY =1000, ΣX² = 200, ΣY² = 5642.

• (i)) Determine the correlation coefficient between X and Y. (150 words)
• (ii)) Given X = 10, what will be the predicted value of Y? (150 words)
• (iii)) Given Y = 15, what will be the predicted value of X ? (150 words)

Q4. A die is thrown 60 times with the following results: Face of die: 1, 2, 3, 4, 5, 6; Frequency: 8, 7, 12, 8, 14, 11. Test that the die is unbiased at 5% level of significance. Given that at 5, 6 and 7 d.f. the value of x² are 11.070, 15.592 and 14.067 respectively.

Q5. Consider the joint probability density function f(x, y) = y²e⁻ʸ⁽ˣ⁺¹⁾; x ≥ 0, y ≥ 0. Are both x and y regressions linear? Give reasons for your answer.

Q6. Solve the following parts.

• (a)) The mean and standard deviation of a variable x are μ and σ respectively. Obtain the mean and standard deviation of (ax+b)/c, where a, b and c are constants. (200 words)
• (b)) If X is a random variable such that E(X)=3 and E(X²)=13, determine a lower bound for P(−2 < X < 8). (200 words)

Q7. Solve the following parts.

• (a)) Let E₁, E₂, E₃ and E₄ be arbitrary events. Write the following events in set notations: i) not more than one of E₁, E₂, E₃, E₄. ii) one and only one of E₁, E₂, E₃, E₄. iii) E₁ and at least one of E₂, E₃, E₄. iv) none of E₂, E₃ and E₄ using E₁. (180 words)
• (b)) Let the probability density function of r.v. X be f(x)= [1+x; −1<x ≤ 0, [1-x; 0<x<1, [0; otherwise and if u = X and v = X², find Cov(u,v). Also check the independence of u and v. (270 words)

Q8. Solve the following parts.

• (a)) For a mesokurtic distribution with standard deviation 5, find fourth central moment m₄. (180 words)
• (b)) The probability that a card will have a flat tyre while crossing a certain bridge is 0.00005. Find the probability that, among 10,000 cars crossing the bridge, i) exactly two cars will have a flat tyre. ii) at most two cards will have a flat tyre. (270 words)

Q9. Solve the following part.

• (a)) Let X₁ be an observation from an exponential distribution with the p.d.f. f(x)=1/θ * e^(-x/θ); x > 0 Test the null hypothesis that the mean of the distribution is θ=2 against the alternative hypothesis that is θ=5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. Find the probabilities of type-I and type-II errors. (200 words)

Q10. Solve the following parts.

• (a)) Let X be a gamma variable with parameters α and λ, having E(X) = 6 and Var (X) = 3. Find α and λ. Also, find the m.g.f. of a gamma variable, and hence verify that mean of X is 6 and variance of X is 3 using m.g.f. (270 words)
• (b)) For married couples living in a certain locality, the probability that the husband will vote in a school board election is 0.21, the probability that they both will vote is 0.15. What is the probability that i) at least one of them will vote? ii) neither of them will vote? (180 words)