# Latex form code

*label*Mathematics

*timer*Asked: Feb 24th, 2019

*account_balance_wallet*$10

### Question Description

the notes is written by hand from stat class, so I need some one know how to write in latex.

## Tutor Answer

Hello Buddy! Here is my final document.Feel free to ask for any edits asap .Charles

Discrete Random Variable

LetY be a random variable

Ry ≡ Range of Y

≡ set of all possible values of Y

≡ finite or countable(countably infinite)

The p.m.f(probability mass function) of Y

if Ry = y1 , y2 , . . . , yk i.e Ry is finite.

Y

y1

y2

y3

PY (y)

,

,

PY (y3 )

PY (y) = P (Y = y)

...

yk

...

PY (yk )

they must have

(a)PY (y) ≥ 0

(b)

k

X

PY (yk ) = 1

j=1

Example 1

Discrete uniform on counting number

Y ∼ uniform(1,. . .,n)

Y follows a discrete uniform distribution on 1, 2, . . . , n

step 1→ RY = [1, 2, . . . , n]

step 2 → pmf of Y is given by;

PY (y) =

1

n

∀y ∈ 1, . . . , n

Y

1

2

...

n

PY (y)

1

n

1

n

...

1

n

Example 2

RY = [−2, −1, 0, 3, 7]

PY (y) = (y 2 + 1)c

Find c such that PY (y) is a valid p.m.f.

1

solution

PY (y)=c[5+2+1+10+48]

c[33]=1

1

33

c=

Pk

j=1

PY (yk )=1

=PY (−2) + PY (−1) + PY (0) + PY (3) + PY (7)=1

c(68)=1

1

c= 68

RY = [−2, −1, 0, 3, 7]

PY (y) = P (Y = y)

1

= 68

(y 2 + 1)

PY (y) = P (Y = y)=

1 (y 2 + 1), if y ∈ Ry

68

0,

1

otherwise.

Example 3

let y=

1

if outcomeis”success”

o

if outcomeis”f ailure”

=1 (w is ”success”)

RY = [0, 1]

PY (y) = P (Y = y) = P y (1 − P )1−y

where P=P(w is ”success”)

Y

0

1

PY (y)

1-P

P

Example 4

Binomial Distribution

Y ≡ Number of ”success” out of n independent Bernoulli trials.

RY = [0, 1, 2, 3, . . . , n]

Pn

Y = i=1 λ1(wi is ”success”)

pmf of Y, PY (y) = P(Y = y) =

n!

Py (1 − P)n−y

y!(n − y)!

2

Cumulative Distribution(C.D.F)

Let Y be a discrete random variable on RY with pmf

Y

(y)

then the cdf of Y is given by;

FY (y) = P(Y ≤ y) = PY (y1 ) + PY (y2 ) + . . . + PY (yn ) =

a≤y

X

PY (a)

Example 5

Y ∼ Bin(3, 21 )

write down;

(a) RY = [0, 1, 2, 3]

(b) PY (y) =

3!

3!

( 1 )y ( 1 )3−y =

( 1 )3

y!(3 − y)! 2 2

y!(3 − y)! 2

(c) FY (y) is table format

(d) FY (y) in compact algebraic form

Solution

(a)

(c)

Y

0

1

2

3

PY (y)

1

8

3

8

3

8

1

8

Y

0

1

2

3

FY (y)

1

8

1

2

7

8

1

(d) FY (y) =

P

a≤y

PY (a)

* Given FY (y), it is often very desired to obtain quantiles of inverse cdf numbers

*Definition

Given α ∈ (0, 1]

FY−1 (α) = inf[y ∈ RY : FY (y) ≡ P(Y ≤ y) = α]

If Y ∼ Bin(3, 21 )

then FY−1 ( 87 ) = 2

3

Question

Fi...

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