Probability Mass Function & Central Limit Theorem: Discrete Variables & Independent Events, Slides of Engineering Mathematics

Download Probability Mass Function & Central Limit Theorem: Discrete Variables & Independent Events and more Slides Engineering Mathematics in PDF only on Docsity!

Probability mass function

and central limit theorem

Review: What is a function?

Domain (^) Range

x

y y = f(x)

s t

t = f(s)

w y = f(w)

Example: Linear transformation

• A linear transformation given by T( x,y ) = ( x +0.25 y , y )

x  x + 0.25 y y  y

Linear transformation maps a parallelogram to a parallelogram

Eigenvectors are vectors whose direction does not change after transformation, T( v ) =  v.

Domain

Range

Example from ENGG

Linear time- invariant system

s(t) (^) T( s(t) )

(an RLC circuit for example)

T

Sinusoidal signals are eigen-functions. Input is sinusoidal  Output is sinusoidal

Domain Range

Periodic signal Periodic signal

Example

• Pick a random point in a rectangular board

• You win

• $2 if the point is inside the triangle.
• $1 if the point is insider the circle.
• nothing otherwise.

• Let X( p ) be the winnings

as a function of the

random point p.

• X( p ) = 2 if p is inside the

triangle.

• X( p ) = 1 if p is inside the

circle

• X( p ) = 0 if p is outside the

triangle and the circle.

Real-life example: lucky rainbow

Probability mass function

• Motivation: Sometimes, the

underlying sample space 

is very complicated.

• We may try to forget the

sample space and work with

the probability mass

function (pmf),

f ( i ) = Pr(X() = i ).

• If we want to emphasize

that it is the pmf of random

variable X(), we can write

fX ( i ).

i

X()

f(i) = Pr( )

Example

• Random experiment: toss n fair coins.

• The sample space  contains 2 n^ outcomes, and the

outcomes are equally likely.

• An outcome  is a string of H and T of length n.

• Let X() be the number of heads in .

• Let Y() be the number of tails in .

• As functions, X() is not equal to Y(). For example:

• X(HHHTHHT) = 5.
• Y(HHHTHHT) = 2. // same input, different outputs

• But X() and Y() have the same probability mass function.

• For i =0,1,…,n,

Identically distributed RV

• Two random variables X and Y, whose underlying

sample spaces are not necessarily the same, are

said to be identically distributed if

f X( i ) = f Y( i ) for all i.

• The example in the previous slide is an example

of identically distributed random variables.

• By looking at the pmf’s of two identically

distributed random variables, we cannot tell

whether the sample spaces behind them are the

same or not.

Independent random variables

• Two discrete random variables X()

and Y() defined on the same sample

space  are said to be statistically

independent if

Pr(X() = i and Y() = j ) = fX( i ) fY( j ) for

all i and j.

• Example: Throw an isocahedral die

and a dodecahedral die at the same

time. The values of the two dice are

independent (but not identically

distributed).

Indepedent vs identically distributed

• In each of the four examples, we pick a point randomly in the area.

X=1, Y=

X=2, Y=

X=1, Y=

X=2, Y=

X and Y are independent and i.d.

X=1, Y=3 X=2, Y=2 X=3, Y=

X and Y are i.d. but not independent

X=1, Y=

X=2, Y=

X=1, Y=

X=2, Y=

X=1, Y=

X=2, Y=

X and Y are independent but not i.d.

X=1, Y=

X=2, Y=

X=1, Y=

X=2, Y=

X and Y are not i.d. and not independent. docsity.com

Rules of the game

• Toss a fair coin repeatedly until we get a head.

• I give you

– $0 if we have a head in the first toss

– $1 if we have a head in the second toss

– $2 if we have a head in the third toss

• In other words, the amount of money I give

you is equal to the number of tails.

Tree diagram

• Let X be the amount I give

you at the end of the

game.

• The pmf of X is

for i =0,1,2,3,4,…

T^ H

H

H

T

T

T^ H

Question

• This is not a free game.

• One need to pay c dollars to play this coin-

tossing game.

• Suppose that I am the host of the game and I

play this game with 1000 people. Each of

them pay $c. What should be the value of c if

I do not want to lose money?

Method of simulation

• Most computer language comes with a

pseudo-random number generator.

• The number returned can be regarded as

uniformly distributed between 0 and 1.

• Construct another random variable Y, which is

identically distributed to X, so that Y can be

generated by computer easily.

• Then we can collect statistics from r.v. Y and

get some intuition.