Question

2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1 <r< 1 and -1 <y<1, and 0 otherwise. Here is a constant. (a) Determine the value of C. (b) Are X and Y independent? (Explain why or why not.) (c) Calculate the probability that 2X - Y > 0 (d) Calculate the probability that |X+Y| < 2 3. Suppose that X1 and X2 are independent and each is standard uniform on (0,1]. Let Y = max{X1, X2}. Calculate E[Y]. (Hint: first work out the CDF of Y, then the PDF.)

Answer 1

2) The given joint PDF is

f(x,y) = C, -1<x<1,-1<y <1

a) The condition for PDF is

| f(x,y)dydx = 1 Cdydx = 1 -1 1 -

b) The marginal PDFs are

fx(x) = [ f(x,y)ay fx(x) = { cay fx(x) = 20 fx(x) = -1<x<1

Similarly,

fr(Y) = 3,-1<y<1

Since , X and Y are independent.

f(x,y) = fylyfx(x)

c) The region is shaded below.

2X-Y>0

(0.5, (1,1) y = 2x 2X - Y >0 (-1,-1) (-0.5,-1)
The area of the shaded region is
2 +0.25-0.25=3
. The total area is 4.

The probability,

P (2X - Y>0) = 2/4 = 0.5

d) The region
X + Y <2.
is equivalent to
-2 5X + Y = 2
. The support is included in the region
X + Y <2.
.

Hence

P(X + Y <2) = 4/4 = 1