Docsity
Log inSign up
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources
Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study
Go to the blog

Stochastic Independence and Functions of Random Variables, Study notes of Mathematical Statistics

Bhupendra Narayan Mandal UniversityMathematical Statistics

The concept of stochastic independence of random variables and how to determine if two random variables are stochastically independent. It also covers the concept of functions of random variables, including how to find the probability density function (pdf) of a function of a random variable. Examples are provided to illustrate these concepts.

Typology: Study notes

2012/2013

Uploaded on 04/20/2013

sathyanarayana
sathyanarayana 🇮🇳

4.4

(21)

140 documents

1 / 54

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
3%
Independent%Random%Variables%
%
• Intuitively, the rvs X and Y are independent
if the distribution of one rv does not in any
way influence distribution of the other rv.
• Independence is a useful assumption for
hydrologic analysis in many situations.
However, there must be a sound physical
basis for the assumption.
Docsity.com
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
pf1f
pf20
pf21
pf22
pf23
pf24
pf25
pf26
pf27
pf28
pf29
pf2a
pf2b
pf2c
pf2d
pf2e
pf2f
pf30
pf31
pf32
pf33
pf34
pf35
pf36

Partial preview of the text

Download Stochastic Independence and Functions of Random Variables and more Study notes Mathematical Statistics in PDF only on Docsity!

3

Independent Random Variables

  • Intuitively, the rvs X and Y are independent

if the distribution of one rv does not in any

way influence distribution of the other rv.

  • Independence is a useful assumption for

hydrologic analysis in many situations.

However, there must be a sound physical

basis for the assumption.

4

  • As an example, inflow to a reservoir (X) and the rainfall

in the command area (Y) may be taken as independent,

if the command area is far removed from the reservoir.

Inflow

Reservoir

Command area

Rainfall

Independent Random Variables

6

Independent Random Variables

  • When two rvs are independent, g(x/y)=g(x)
    • Distribution of X given Y is independent of Y and

hence the conditional pdf is equalt to the marginal pdf.

( )

f x y

g x y

h y

f x y

g x

h y

f x y g x h y

h(y) > 0

77

Independent Random Variables

  • The random variables X and Y are stochastically

independent if and only if their joint density is equal to

the product of their marginal densities.

  • Discrete case: the two r.v.s are independent if and only if

p(x

i

, y

j

) = p(x

i

). p(y

j

) v i,j

Example-1(contd.)

( )

1 1

0 0

1

2

0

f x y dy x y dy
y
xy x x

∫ ∫

g(x)
h(y)

( )

1 1

0 0

1

2

0

f x y dx x y dx
x
xy y y

∫ ∫

9

Example-1(contd.)

g x h y x y

× = + × +

Therefore X and Y are not stochastically
independent.

f ( , x y ) ≠ g x h y ( ). ( )

10

Example-2(contd.)

( )

0 0

0

x y

x y x

f x y dy e dy
e e dy e x

∞ ∞

− +

∞

− − −

∫ ∫

∫

g(x)
h(y)

( )

0 0

0

x y

y x y

f x y dx e dx
e e dx e y

∞ ∞

− +

∞

− − −

∫ ∫

∫

12

Example-2 (contd.)

( )

x y

x y

g x h y e e

e

− −

− +

× = ×

Therefore X and Y are stochastically
independent

f ( , x y ) = g x h y ( ). ( )

13

  • X: discrete; Y = H(X), a funcAon of X.
  • The pmf of X is known
  • Enumerate possible values of Y for the

discrete values of X

  • Then obtain the probabiliAes of the possible

values of Y from the probabiliAes of the

corresponding values of X.

Functions of Random Variable

Example for discrete case

p(x) = 60/77x ; x = 2, 3, 4, 5

discrete values

y = x

2

-7x+

Distribution of y:

p(Y=-7) = p(X=3)+p(X=4) = 20/77+ 15/77 = 35/77 = 5/

p(Y=-5) = p(X=2)+p(X=5) = 30/77+ 12/77 = 42/77 = 6/

x 2 3 4 5

y -­‐5 -­‐7 -­‐7 -­‐

p(x) 30/77 20/77 15/77 12/

16

b. Differentiate G(y) w.r.t y to get g(y)
c. Since g(y) must be non-negative, determine
those values of y over which g(y) > 0
and check,

General procedure for functions of continuous

random variables:

g y dy ( ) 1

∞

−∞

18

The rv X has a pdf
f(x) = x/2 0 < x < 2
= 0 elsewhere
Let H(X) = 4X+
Find the pdf of Y=H(X)
a. Get the CDF of Y
G(y) = P [Y < y]
= P [4X+1 < y]

1

4

y

P X

−

⎡ ⎤

≤

⎢ ⎥

⎣ ⎦

Example-

19

b. g(y) =

c. From 0 < x < 1, we get

( ) ( )

2

1

64

dG y y

d

dy dy

⎛ ⎞

−

⎜ ⎟

⎜ ⎟

⎝ ⎠

( )

1

( ) 1 9

32

y

g y y

−

= < <

Example-1(contd.)

( )

y
y

21

y=4x+

Example-1(contd.)

Check :

9

1

y

dy

9

9

2

1

1

2

( 1) 1 ( 1)

32 32 2

1

(8 0)

64

1

y y

dy

⎡ ⎤

− −

=

⎢ ⎥

⎣ ⎦

= −

=

22