Which of the following statements are True or False? Give short proof

Question

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-1and-3Y-lis-0.48.

ii) If X and Y are independent binomial variates with parameters (n₁, p₁) and (n₂, p₂) respectively, then X + Y has binomial distribution with parameters (n₁+n₂, p₁ + p₂).

iii) The function defined as

$f(x)=\begin{cases}|x|,&-1<x<1\\0,&\text{otherwise}\end{cases}$

is a probability density function.

iv) For a normal distribution with mean μand variance σ², the hypotheses

Η₁: μ = μο, σ² = 1 and

Η₂: μ =μο, σ² ≥1 are simple hypotheses.

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.

18 Feb 2025

Answer :

Word Count : 684

Let's analyze each statement one by one. ### Statement (i): *If the correlation coefficient between $X$ and $Y$ is 0.8, then the correlation coefficient between $2X-1$ and $-3Y-1$ is $-0.48$.* #### Solution: The correlation coefficient is unchanged under linear transformations of the form $ aX + b $ where $ a > 0 $, but if $ a < 0 $, it flips the sign. The formula for the correlation coefficient after linear transformation is: \[ \rho_{aX+b, cY+d} = \text{sign}(a) \cdot \text{sign}(c) \cdot \rho_{X,Y} \] For $ 2X-1 $ and $ -3Y-1 $, we have: - $ a = 2 $ (positive, so sign remains $ +1 $) - $ c = -3 $ (negative, so sign flips $ -1 $) - Given $ \rho_{X,Y} = 0.8 $, we get: \[ \rho_{(2X-1), (-3Y-1)} = (1)(-1)(0.8) = __________ ______ ____ _________ ___ ________ ____ ___ _______.
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