Probability and Regression Analysis

This study material covers the topics of probability and regression analysis. It discusses the independence of events, correlation between variables, and the impact of firm profitability on CEO salaries. It also explores the concept of slopes and regression models.

Probability and Regression Analysis
Part 1
Question 1
(a). Probability (have a second child) = 0.684
(b). Probability (have a second | 1st child a boy) = 0.326
(c). Probability (have a second | 1st child a girl) = 0.358
(d). Yes, the events “have a second child) and “sex of first child” are independent events simply because one
event does not affect the outcome of the other. Regardless of whether the couple had a boy or girl the first time
the decision to have a second child is entirely influenced by the parents.
Question 2
(a). No, the two variables X and Y are not independent simply because X can be computed using a function of
Y
(b). No, the two variables X and Y are correlated i.e.
The expected value of the product of two independent variables is the product of expected value of those two
variables. That’s not the case here.
E ( X . Y ) =E ( X ) . E(Y )
E ( X . Y ) =E ( Y 2 ) . E(Y )
E ( X . Y )=E ( Y 3 )
E ( X . Y )=E ( Y 3 ) =0
In addition their covariance is also 0 i.e. cov(X,Y)=0
(c). Uncorrelated variables are independent. FALSE
(d). Independent variables are uncorrelated. TRUE
Question 3
The slopes are not reciprocals
With regard to basic statistical representation of the aforementioned problem we can see that
Y =ax+b where a isthe slope

X =cy +d where c is the slope
It is obvious that “a” is not equivalent to “c”, nor is “c” the reciprocal of “a”. This is due to fact that
interchanging the dependent and independent variable completely alters the resultant regression model. This can
be clearly demonstrated on a scatter plot where the data points are t
Question 4
(a) . Y = β0 +β1 X
Where Y=CEO Salary and X=Millions of Dollars of Profit
(b). yes, you can determine the average salary based on the information given
Y =476.92+0.572( X )
X=200
Y =476.92+0.572∗200
Y =476.92+114.4
Y =591.32
Where Y is the average salary of CEO in thousands of dollars;
Average CEO Salary= $591,320
(c).
(i). If the company breaks even then the profit will be zero as such the value of X will also be zero. Then the
salary of the CEO is given to be
Y =476.92+0.572( X )
X=0
Y =476.92+0.572∗0
Y =476.92
CEO Salary= $476,920
(ii). For every additional million increment in profit we expect the salary of the CEO to increase by a given
figure. The constant figure of the CEO’s salary is given by 476.92 in the model. While, the variable figure of
the CEO’s salary is given by 0.572(X); where X is the increment or decrement in profit.
0.572 ( x )=Change∈CEO salary for every X millionincrement ∈Profit

0.572 ( 1 )=Change∈CEO salary for every million increment ∈Profit
$ 572=Change∈CEO salary for every million increment ∈Profit
(iii). A loss of 100 million dollars would result in a reduction to the CEO’s salary when compared to a situation
where the company made not profit or loss.
Y =476.92+0.572( X )
X=-100
Y =476.92+0.572∗−100
Y =476.92−57.2
Y =419.7 2
CEO Salary= $419,720
(d). No, simply because there may be other factors besides firm profitability that affect the salary payable to a
CEO. 100% of the change in the CEO’s salary cannot be explained by the profits made by the firm.
(e). The coefficient will be biased because it will cause considerable change in the CEO’s salary than it should
if the variable firm size was included. Consider the hypothesized value of the coefficient for profits indicated
below:
Y =476.92+0.384 ( firm profit ) +4 0.57 ¿