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. (80 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). (80 words)
- (iii)) The function defined as f(x) = [|x| ;-1<x<1; 0; otherwise is a probability density function. (80 words)
- (iv)) For a normal distribution with mean uand variance σ², the hypotheses Η₁ : μ = μο, σ² = 1and H₂ : μ = μο, σ² ≥ 1 are simple hypotheses. (80 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. (80 words)
Key Points:
- Correlation coefficient: ρ(aX+b, cY+d) = sign(ac) * ρ(X, Y).
- Sum of independent binomials is binomial only if probabilities of success are identical.
- A function is a PDF if f(x) ≥ 0 and ∫f(x)dx = 1.
- Simple hypothesis: Completely specifies all parameters of the distribution.
Answer: This response addresses five statements related to probability and statistics concepts, including correlation, distribution properties, probability density functions, hypothesis types, and power of a test. Each statement is evaluated as True or False, accompanied by a short proof or counterexample, drawing upon fundamental definitions and theorems from the MTE-11 course material. Adherence to specific word limits for each sub-question is maintained, ensuring concise yet comprehensive explanation...