Problems under continuous random variables

PROBLEMS UNDER CONTINUOUS RANDOM VARIABLES

Example 2.1.9

Suppose the point Probability Density Function (PDF) is given by

Obtain the marginal PDF of X and that of Y. Hence, otherwise find P [1/4 ≤ y ≤ 3/4]

Solution:

 

Example 2.1.10

Let X and Y have j.p.d.f f(x, y) = 2, 0 < x < y < 1. Find the m.d.f. and the conditional density function of Y given X = x.  [A.U. A/M 2003] [A.U CBT A/M 2011]

Solution:

The marginal density function of X is given by

 

Example 2.1.11

The joint probability density function of a random variable X and Y is given by,

Find the marginal densities of X and Y.

Also, prove that X and Y are independent.

Solution:

The marginal density of X is given by,

 

Example 2.1.12

The joint p.d.f of the random variable (X, Y) is given by f(x, y) = Kxy e-^{(x2 + y2)}, x > 0, y > 0. Find the value of K and also prove that X and Y are independent.

 [A.U. May, 2000, 2004, N/D 2006, N/D 2011, M/J 2012] [N/D 2007, M/J 2009, Tvli A/M 2009, N/D 2013] [A.U A/M 2015 (RP) R13, R8] [A.U N/D 2018 R-13 RP]

Solution: Here, the range space is the entire first quadrant of the xy-plane.

 

Example 2.1.13

Given fxy (x, y) = Cx (x − y), 0 < x < 2, -x < y < x and 0, elsewhere

 (a) Evaluate C; (b) Find f_x(x); (c) f_y/x (y/x). and (d) fy (y)

[A.U. N/D 2004, M/J 2006, N/D 2010, M/J 2013] [A.U N/D 2016 R13 (PQT, RP)]

Solution : By the property of j.p.d.f, we have,

 

Example 2.1.14

Suppose that X and Y are independent and that these are the distribution tables for X and Y.

What is the joint probability space?

Solution: Since, X and Y are independent,

f(x, y) = f(x). f(y), ∀ (x,y)

Hence, the joint probability space is given by,

 

Example 2.1.15

If f(x, y) = K (1 − x − y), 0 < x & y < 1/2, is a joint density function, then find K. [A.U Tvli M/J 2011]

Solution: By the property of joint p.d.f, we have 

Example 2.1.16

If the joint p.d.f of (X, Y) is f(x, y) = 6e^-2x-3y, x = 0, y ≥ 0, find the marginal density of X and conditional density of Y given X.

Solution: The marginal density of X is given by

 

Example 2.1.17

The j.p.d.f of (X, Y) is given by f(x,y) = e^{−(x + y)}, 0 ≤ x, y < ∞.

Are X and Y independent? Why?

[A.U. A/M.2008] [A.U Tvli M/J 2010, Trichy M/J 2011, N/D 2011] [A.U N/D 2015 R13 RP] [A.U A/M 2017 R-13]

Solution:

Given: f (x, y) = e^{-(x + y)}, 0 ≤  x, y < ∞

To prove: X and Y are independent.

i.e., To Prove: f (x)f (v) = f(x, y)

Hence, X and Y are independent.

 

Example 2.1.18

If the joint pdf of a two-dimensional random variable (X, Y) is given by,

f(x,y) = x2 + xy/3, 0 < x < 1;0 < y < 2

= 0 , elsewhere

Find (i) P(X > 1/2) ; (ii) P(Y < X) and (iii) P (Y < 1/2 / X < 1/2)

Check whether the conditional density functions are valid.

[A.U Trichy A/M 2010] [A.U A/M 2011, M/J 2009, M/J 2014]

[A.U A/M 2015 (RP) R8] [A.U N/D 2019 (R-17) PS]

Solution: Given: 0 < x < 1, 0 < y < 2

 

Example 2.1.19

Given that the joint p.d.f of (X, Y) is

f(x, y) = e-y, x > 0, y > x

= 0, elsewhere.

Find (i) P (X > 1/Y < 5) and

(ii) the marginal distributions of X and Y

Solution: Given: x > 0, y > x

Example 2.1.20

The joint density function of the RVS X and Y is given by,

f(x, y) = 8xy, 0 < x < 1; 0 < y < x

= 0, elsewhere

[A.U Trichy A/M 2010]

Find P(Y < 1/8/ X < 1/2). Also find the conditional density function of

[AU, May 1999, N/D 2005, N/D 2009]

Solution :

The marginal density function of X is given by,

 

Example 2.1.21

If the joint p.d.f of (X, Y) is given by f(x, y) = K, 0 ≤x≤ y ≤2 find K and also the marginal and conditional density functions.

Solution :

By the property of joint p.d.f we have

 

Example 2.1.22

If the joint density function of the two random variables 'X' and 'Y' be

f(x, y) = e^{-(x + y)}, x ≥ 0, y ≥ 0

= 0, otherwise

 [A.U N/D. 2003]

Find (i) P(X < 1) and (ii) P(X + Y < 1)

 [A.U N/D. 2009] [A.U CBT M/J 2010] [A.U N/D 2019 (R17) PQT]

Solution :

 To find the marginal density function of X.

Let the marginal density function of X be g (x) and it is defined as

 

Example 2.1.23

The joint p.d.f of the random variables X and Y is given by

P(x, y) = xe ^{-x (y+1)} where 0 ≤ x, y < ∞. (i) Find P(x) and P(y) and

(ii) Are the random variables independent ?

Solution:

 

Example 2.1.24

S.T. the function

Solution :

(i) f(x, y) ≥ 0 in the given interval, 0 ≤ x, y ≤ 1

 

Example 2.1.25

If the joint distribution function of X and Y is given by

F(x, y) = (1 – e^-x) (1 − e^-y),

for x > 0, y > 0

otherwise

(i) Find the marginal density of X and Y

(ii) Are X and Y independent ?

(iii) P(1 < X < 3, 1 < Y < 2)

Solution:

 

Example 2.1.26

The joint density function of two random variables X and Y is

 

Example 2.1.29

 

Example 2.1.28

Given the joint p.d.f of (X, Y) as of

Find the marginal and conditional p.d.f of X and Y. Are X and Y independent ?

[A.U Tvli M/J 2010] [A.U A/M 2017 R-13]

[A.U N/D 2017 R-13] [A.U A/M 2019 (R17) PS]

Solution:

 

Example 2.1.29

The joint p.d.f of two random variables X and Y is given by

Find the marginal distributions of X and Y, the conditional distribution of Y for X = x and the expected value of this conditional distribution.

[A.U Trichy M/J 2009] [A.U. A/M 2004, A/M 2011]

Solution: (i) The marginal distribution of X is

From the form of the joint p.d.f of (X, Y). Similarly,

The marginal distribution of Y is

 

Example 2.1.30

Find k if the joint probability density function of a bivariate r.v.

(X, Y) is given by f(x, y) =

 [AU Dec. 2006, A/M 2008, Tvli A/M 2009, A.U A/M 2010, M/J 2014] [A.U N/D 2017 (RP) R-13]

Solution: If f (x, y) is a joint p.d.f, then