State which of the following statements are true and which are false.

Question

State which of the following statements are true and which are false. Give a short proof or a counter example in support of your answer.

a) For three independent events E₁ , E₂and E₃,

$P(E_1\cup E_2 \cup E_3) + P(\bar{E_1})P(\bar{E_2})P(\bar{E_3}) = 0$

b) The range of multiple and partial correlation coefficient is $]-1,1[$
c) If $\left \{ X(t); t \geq 0 \right \}$ is a poisson process, then $N(t) = [X(t + S_0) - X(t)]$ where $S_0 > 0$ is a
fixed constant, is also a poisson process.
d) In Hotelling T² , the value of S is given by

$S = \frac{1}{n-1}\sum_{j=1}^{n}(X_j - \mu)(X_j - \mu)'$
e) Let $X_{p\times 1} \sim N_p(\mu,\Sigma)$ and $X_{p\times n}$
be the state matrix, then parameters involved in the above
distribution are p for $\mu$ and $\frac{1}{2}p(p+1)$ for $\Sigma$

06 Feb 2024

Answer :

Word Count : 272

a) False. Let's consider a counterexample where each event has a probability of 0.5. Then,
\[ P(E_1 \cup E_2 \cup E_3) + P(\bar{E_1})P(\bar{E_2})P(\bar{E_3}) = 1 + (0.5)^3 = 1.125 \neq 0 \]
So, the statement is false.

b) True. The range of the multiple and partial correlation coefficient is indeed $(-1, 1)$. ___ ________ _________ ______ ______ _____ ________.
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