MSO 201A / ESO 209: Probability and Statistics

2014-2015-II Semester

Assignment-IVInstructor: Neeraj Misra

1. Let

F (x; y) =

(1; if x+ 2y 10; if x+ 2y < 1

:

Does F (; ) dene a d.f.?

2. Let

F (x; y) =

(0; if x < 0 or x+ y < 1 or y < 0

1; otherwise:

Does F () dene a d.f.?

3. Let X = (X1; X2) be a bivariate random vector having the d.f.

F (x; y) =

8>>>>>>>>>>>:

0; if x < 0 or y < 01+xy2

; if 0 x < 1; 0 y < 11+x2; if 0 x < 1; y 1

1+y2; if x 1; 0 y < 1

1 if x 1; y 1

:

(a) Verify that F is a d.f.; (b) Determine whether X is of discrete or of AC type; (c)

Find the marginal distribution functions of X1 and X2; (d) Find P (12 X1 1; 14 >>>>>>>>>:

0; if x < 0 or y < 1y216

; if 0 x < 1; 1 y < 212; if 0 x < 1; y 2

y213

; if x 1; 1 y < 21 if x 1; y 2

:

(a) Verify that F is a d.f.; (b) Determine whether X is of discrete or of AC type; (c)

Find the marginal distribution functions of X1 and X2; (d) Find P (12 X1 1; 54