Question

Q3. a. If a random variable X has mean 3 and standard deviation 5, then find out the variance of a variable

Y=2X-5.

b. If each observation in the set of values (X, Y) is divided by 100, then find the correlation coefficient of

new data set?

Answer 1

a. We use the linearity of a mean of random variables and receive: $E[Y]=E[2X-5]=2E[X]-E[5]=2E[X]-5=6-5=1$.

We remind that the standard deviation squared is: $\sigma_Y^2=E[(Y-E[Y])^2]=E[Y^2-2YE[Y]+(E[Y])^2]=E[Y^2]-2(E[Y])^2+(E[Y])^2=E[Y^2]-(E[Y])^2=E[4X^2-20X+25]-1=4E[X^2]-20E[X]+24=4E[X^2]+24-20\cdot3=4E[X^2]+24-60=4E[X^2]-36$

We find $E[X^2]$ from equality: $25=\sigma_X^2=E[X^2]-(E[X])^2=E[X^2]-9$. We receive: $E[X^2]=34$. Finally, we get: $\sigma_Y^2=4\cdot34-36=100$ and $\sigma_Y=10$.

Thus, the mean is 1 and the standard deviation is 10.

b. We remind that the formula for Pearson's correlation coefficient(when applied to sample) is:

$r_{xy}=\frac{\sum_{i=0}^n(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=0}^n(x_i-\bar{x})^2}\sqrt{\sum_{i=0}^n(y_i-\bar{y})^2}}$. As we can see from the formula, the coefficient will remain unchanged.