Question

Which of the following statements are true or false? Give a short proof or a counter
example in support of your answer.
i) If the correlation coefficient between x and y is 0.75, then the correlation
coefficient between (2 + 5x) and (−2y + 3)is − 0.75 .
ii) If P(A) = 0.5, P(A ∪ B) = 0.7 and A and Bare independent events, then 5
2 P(B) = .
iii) Let X1 X2 ,X3 ,..., Xn be a random sample of size n from N(0,σ² ). Then S² =∑(X²/σ²)
follows normal distribution.
iv) A maximum likelihood estimator is always unbiased.
v) The mean deviation is least when deviations are taken about the mean.

Answer 1

i) $\sigma (cX) =|c| \sigma (X), \sigma (X+c) =\sigma (X)$

$\operatorname{cov}(aX,bY) =ab \operatorname{cov}(X,Y), \operatorname{cov}(X+a,Y+b) =\operatorname{cov}(X,Y)$

$\operatorname{corr}(X,Y) =\dfrac{ \operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} \Rightarrow \operatorname{corr}(5X+2,-2Y+3)=\dfrac{-10}{10}\operatorname{corr}(X,Y)=-0.75$

True

ii) $P(A\cup B)=P(A)+P(B)-P(A\cap B), P(A\cap B)=P(A)P(B)$

$P(B)=\dfrac{P(A\cup B) - P(A)}{1-P(A)}=\dfrac{0.2}{0.5}=0.4$

iii) False.

If $X_k \sim N(0,\sigma^2 ), k=1,\dots, n,$ then

$$\dfrac{1}{\sigma^2}\sum\limits_{k=1}^n X_k^2 \sim \chi^2(n)$$

https://en.wikipedia.org/wiki/Chi-squared_distribution

iv) False

https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Higher-order_properties

see bias of order $1/n$

v) False

https://www.emathzone.com/tutorials/basic-statistics/mean-deviation-and-its-coefficient.html

see median case