1. Which of the following is true?
1. If the correlation coefficient is zero, the variables are independent variables
2. If the variables X and Y are related as Y = e^2x, then if X is normally distributed, then Y^2 is log normally distributed
3. X and Y = Ln(x) would have correlation coefficient as 1
4. If the variables are independent, then their correlation coefficient is 1

Solution:
A is not correct. For example, if the relation is harmonic, then the correlation can be zero by they are clearly not independent
D is not correct,
C Correlation coefficient is 1 only for linear relationships

2. To check the signifiance of regression variables, you would consider which of the following with highest priority
1. p-value of regression variable
2. R-Square value
3. R value
4. Standard Error of Regression Variable

Solution:
If we are looking for significance of variables, then p-value is the most critical factor.

3. I dont know the standard deviation and mean of my population – The average salary of people in my colony. I want to launch a new premium service which is targeted towards people with income > \$100,000 per year. I check on 25 people in my colony and find their average income to be \$95,000 and variance to be \$14,000. What statistical test should I perform to launch my product?

1. Z (Normal) Test checking for mean income to be greater than \$100,000
2. Chi-Square Test with 24 degrees of freedom checking for volatility to be greater than \$14,000
3. T-Test with 24 degrees of freedom checking for mean income to be greater than \$100,000
4. F-Test with 24 Degrees of freedom checking for volatility to be greater than \$14,000

Solution:

4. If P(A) = a; P(B) = b, then which of the following is true:

1. p(A/B) <= (a+b-1)/b
2. P(A or B) = a + b – a*b
3. P(A and B) = a * b
4. P(A/B) >= (a+b-1)/b

Solution:
Since we are testing for means and volatility is unknown and n < 30, so we need to use the T-Test 4) D P(A/B) >= (a+b-1)/b is correct.
We know, for any general events A & B,
1) P(A/B) = P(A and B)/P(B) so P(A & B) = P(A/B) * b
We also know that:
2) P(A or B) = P(A) + P(B) – P(A and B) and
3) P(A or B) <= 1 (because probability <=1) => P(A or B) = a + b – P(A/B) * b (from 1 & 2)
so a + b – P(A/B) * b <=1 (from 3) so P(A/B) >= (a+b-1)/b

5. The random variable x with the following probability density function:

f(x) = 1/(b – a) for a < x < b

= 0 otherwise
is said to be a uniformly distributed over (a, b). Calculate its variance.

1. (a-b)2/4
2. (a+b)/2
3. (b-a)/4
4. (b-a)2/12

Solution:
Mean (Âµ) of uniform distribution = (a+b)/2
Variance = S(x-Âµ)2 /n
= S{x-(a+b)/2}2/b-a

As it’s a continuous distribution we have to integrate it as:
Solving this gives (b-a)2/12.

6. Calculate the probability of a subsidiary and parent company both defaulting over the next year. Assume that the subsidiary will default if the parent defaults, but the parent will not necessarily default if the subsidiary defaults. Also assume that the parent has a one-year probability of default of 0.70% and the subsidiary has a one-year probability of default of 0.80%.

1. 0.7
2. 0.56
3. 0.14
4. 0.24

Solution: