Probability and Statistics

Which one of the following statements are true?
a)  $0\le\gamma\le1$  
b) If  $\gamma=0$  then cov(x,y)=0.

c)  $\gamma=\frac{\left(\sigma_x^2+\sigma_y^2-\left(\sigma_{x-y}\right)^2\right)}{2\sigma_x\sigma_y}$  

d)  $\gamma$  is independent of change of origin and scale.

• a, b and c are true

• only c is true

• b and c are true

• a and d are true

The acute angle between the regression lines is given by

• $\tan^{-1}\left[\left(\frac{\gamma^2-1}{\gamma}\right)\left(\frac{\sigma_x\sigma_y}{\sigma_x^2+\sigma_y^2}\right)\right]$

• $\tan^{-1}\left[\left(\frac{\gamma}{\gamma^2-1}\right)\left(\frac{\sigma_x\sigma_y}{\sigma_x^2+\sigma_y^2}\right)\right]$

• $\tan^{-1}\left[\left(\frac{\gamma}{1-\gamma^2}\right)\left(\frac{\sigma_x\sigma_y}{\sigma_x^2+\sigma_y^2}\right)\right]$

• $\tan^{-1}\left[\left(\frac{1-\gamma^2}{\gamma}\right)\left(\frac{\sigma_x\sigma_y}{\sigma_x^2+\sigma_y^2}\right)\right]$

Which one of the following is an ideal index number?

• Bowley's index number

• Fisher's index number

• Edge-Worth's index number

• Paasche's index number

Let A, B, C be any three events in a sample space S associated with a random experiment. If atleast one of the three events happens their related set theoretic expression is given by

• $\overline{A}\ \cap\overline{B}\ \cap\overline{C\ }$

• $A\cap\overline{B}\ \cap\overline{C}$

• $A\cup B\cup C$

• $A\cap B\cap C$

The chance of X solving a particular problem is  $\frac{2}{3}$  and the chance of Y solving the problem is  $\frac{3}{4}$  . What is the probability that the problem is solved if they try independently?

•  $\frac{9}{12}$  

•  $\frac{11}{12}$  

•  $\frac{17}{12}$  

•  $\frac{1}{12}$  

If X is a continuous random variable the probability of every set consisting of a single point is

• $\infty$

• 0

• 1

• $-\infty$

If  $\phi_X\left(t\right)$  is the characteristic function of a random variable X  then  $\phi\left(0\right)$  is

• 0

• 1

• -1

•  $\infty$  

In a binomial distribution the mean is 4 and the variance is  $\frac{8}{3}$  . Then the mode is

• 8

• 3

• 12

• 4

In which of the following distributions mean and variance are the same?

• Chi-Square distribution

• Normal distribution

• Student's t-distribution

• Poisson distribution

For the normal distribution the Quartile deviation, mean deviation and standard deviation are approximately in the ratio

• 12:15:10

• 10:12:15

• 15:12:10

• 10:15:12