Q1. State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
- (a)) If P is a transition matrix of a Markov Chain, then all the rows of lim P^n are identical. (80 words)
- (b)) In a variance-covariance matrix all elements are always positive. (80 words)
- (c)) If X1,X2, X3 are iid from N₂(μ, Σ), then (X1+X2+X3)/3 follows N₂(μ, Σ/3). (80 words)
- (d)) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1. (80 words)
- (e)) For a renewal function M_t, lim_(t->inf) (M_t/t) = 1/μ. (80 words)
Key Points:
- Ergodic Markov chains: lim P^n has identical rows, representing the stationary distribution π.
- Variance-covariance matrix: Diagonal elements (variances) are non-negative; off-diagonal (covariances) can be negative.
- I.i.d. multivariate normal means: (X₁+X₂+X₃)/3 from N₂(μ, Σ) is N₂(μ, Σ/3).
- Partial correlation: Range [-1, 1]; Multiple correlation: Range [0, 1].
Answer: This response addresses five statements related to Probability and Statistics, specifically from the MMT-008 course curriculum. Each statement is evaluated as True or False, followed by a concise justification based on relevant definitions, theorems, or counterexamples from the course material. The topics covered include Markov Chains, multivariate distributions, correlation, and renewal theory.