Homework 3 solutions (4 - UHdblecher/3338H3s.pdfHomework 3 solutions (4.1 and Chapter 3 Q10) Chapter 3 Chapter 4: 6. There are two ways to do these problems. One is to pick your favorite

Homework 3 solutions (4.1 and Chapter 3 Q10)

Chapter 3

Chapter 4:

6. There are two ways to do these problems. One is to pick your favorite discrete r.v. X, and take a = 2 and b = 3 say, draw the graphs of p_X and p_Y, and say what you see. Similarly in (b) , pick your favorite continuous r.v. , and take a = 2 and b = 3 say, draw the graphs of f_X and f_Y, and say what you see.

A second method is to note that in a, p_Y(y) = Pr(a X + b = y) = Pr(X = (y-b)/a) = p_X(y/a – b/a). This means that we are shifting the graph b/a units to the right, and shrinking the graph horizontally by a factor of a. In b, a similar argument shows that F_Y(y) = F_X(y/a – b/a), , at least if a >0. This means that we are shifting the graph b/a units to the right, and shrinking the graph horizontally by a factor of a. Taking the derivative, f_Y(y) = f_X (y/a – b/a)/a. This means that we are shifting the graph b/a units to the right, and shrinking the graph both horizontally and vertically by a factor of a. The answer to c is `slightly different’, as explained above.

8.a)

b)

From a graphical point of view, if you are interested,

9.a

E