State whether the following statements are true or false. Justify your

Question

State whether the following statements are true or false. Justify your answer with a short proof or a counter example:

i) Although in one-step transition probability matrix, ,P of a Markov chain the sum of each row must necessarily be unity but in higher order transition probability matrices, $P^{(j)};j=2,3,4,\dots$ . This rule is not necessary.

ii) For the joint pdf of random variables (X,Y) given by:

$f(x,y)=x^2+\frac{xy}{3},\quad 0\leq x\leq 1,0\leq y\leq 2$

$P(X+Y\geq c)=\frac{65}{72}$

iii) Let the r.v. X follows the negative exponential distribution with parameter $\lambda=\frac{1}{3}$ Then

$P(X>4\mid X>3)=P(X>1)=e^{-\frac{1}{3}}$

iv) One of the examples of non-Poisson queuing system is the (M / G (:)1/ ∞ (FIF0) queuing model.

v) $\text{If}X_1,X_2,\dots,X_n$ be a random sample from $N_p(\mu,\Sigma)$ then maximum likelihood estimators o $\text{}\mu\text{and}\Sigma\text{are:}$

$\hat{\mu}=\overline{X}\quad\text{and}\quad\hat{\Sigma}=\sum_{i=1}^{n}(X_i-\overline{X})(X_i-\overline{X})'$

22 Apr 2025

Answer :

Word Count : 715

Let's examine each statement one by one numerically or conceptually, as requested — without using Python. --- ### Statement (i) Claim: In one-step transition probability matrix $P$ of a Markov chain, the sum of each row is 1, but in higher-order transition probability matrices $P^{(j)}, j=2,3,4,\dots$, this rule is not necessary. Answer: FALSE Justification: Let $P^{(j)} = P^j$, the j-step transition matrix obtained by multiplying the one-step transition matrix $P$ with itself $j$ times. Each entry $p^{(j)}_{ij}$ represents the probability of transitioning from state $i$ to state $j$ in $j$ steps. Since probabilities are preserved over steps, the sum of each row of $P^{(j)}$ must also be 1. Example (2-state Markov Chain): Let $$ P = \begin{bmatrix} 0.4 & 0.6 \\ 0.3 & 0.7 \end{bmatrix} \Rightarrow \text{Row sums: } 1, 1 $$ Now compute $P^2$: $$ P^2 = P \cdot P = \begin{bmatrix} 0.4 & 0.6 \\ 0.3 & 0.7 \end{bmatrix} \cdot \begin{bmatrix} 0.4 & 0.6 \\ 0.3 & 0.7 \end{bmatrix} $$ First row of $P^2$: * (1,1): $0.4 \cdot 0.4 + 0.6 \cdot 0.3 = 0.16 + __________ ________ _______ _____ ________ ________ _________.
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