stats_ch4_notes Discrete Random Variablrs

• stats_Chapter 4.pdf

Discrete Random Variablrs dpqr

Stats_Quiz dpqr2

todo when would I use each one?

Distribution	PMF/PDF (d*)	CDF (p*)	Quantile (q*)	Random (r*)
Normal	dnorm	pnorm	qnorm	rnorm
Binomial	dbinom	pbinom	qbinom	rbinom
Geometric	dgeom	pgeom	qgeom	rgeom
Hypergeometric	dhyper	phyper	qhyper	rhyper
Poisson	dpois	ppois	qpois	rpois

Binom functions

Prob btwn 2 vars of a TRUE/FALSE outcome

• p=prob of susc
• k= # trials

p&dbinom table

• k = goal
• n=$\mu$
• p=$\sigma$

Probability you want	R expression
$P(X\le k)$ (lower tail)	pbinom(k, n, p)
$P(X<k)$	pbinom(k-1, n, p)
$P(X\ge k)$	1 - pbinom(k-1, n, p)
$P(X>k)$	1 - pbinom(k, n, p)
$P(X=k)$	dbinom(k, n, p)
$P(k<X\le m)$	pbinom(m, n, p) - pbinom(k, n, p)

Q&Rbinom

name	expr	r
Quantile function (inverse CDF)	no. fail k s/t $P(x\le k)\ge p$	qbinom(k,n,p)
rbinom	random sample	rbinom(k,n,p)

qbinom(0.95, size=5, prob=0.5)#in 95% of simulations chance you’ll get 5 heads after x runs, in this case 11
dbinom(3, size=5, prob=0.5)   # P(3 tails before 5 heads)
rbinom(1, 10, 0.5)# in x=10 flips, how many heads?

qbinom: P(x failures b4 size-th success) {qbinom}

• Returns failures
• k% of experiments will require at most x{return} failures to reach n successes
• k: $xth$ percentile of trails (95% of trails will have following..)
• n: Size per trial- p: Probability success, eg. x% succeeds (coin flip =0.5)

rbinom

• n= # draws per trial
• size = # of trials
• probability? (coin flip = 0.5)

Other Distribution types

Hypergeometric Distribution

still confused Draw from a finite population without replacement. Eg. Draw 5, total 10red 40 blue

# {d,p,q,r}hyper(...)
dhyper(2, 3, 6, 4)#Probability of drawing 2 red when picking 4 marbles

Normal Distribution

Data = norm distro, takes in mean and sd

xnorm(mean,sd)#where sd = standard deviation
dnorm(x, mean, sd)   # PDF (density)
pnorm(q, mean, sd)   # P(X ≤ q)

Multinomial Distribution

https://chatgpt.com/s/t_68dd61327b648191860af24c2e9eabc6

$$\begin{array}{c}P(y_1=2,y_2=4,y_3=4)\\ \text{n trials}\\ \text{k possible outcomes}\end{array}$$

dmultinom(x = c(2,4,4), prob = c(0.3,0.4,0.3))

Negative Binom

How many trials do I need to perform to get x sucesses? Used to get confidence intervals!

• We generally have to re@param these functions in order to fit into the r function (from qn → this)

Poisson distribution

$$\begin{array}{c}P(Y=y)=\frac{e^{\gamma }{\gamma }^y}{y!}\\ P(y=2,\gamma =4)\\ dplus(2,4)\end{array}$$

To work it out we use:

• d/p/q/rpois
• WE DO NOT USE lower.tail=FALSE
  • Does not include the actual q range (middle valued range), in general it’s NOT what you want
  • If you want a upper tail, use 1-… (see Binom functions

https://chatgpt.com/s/t_68dd6184d2fc8191ac3895ddaed74b21

Used in measuring half-life decay. Lets take some neuclus, 4 particles /s on avg, being released out of it.

Defitions Random Vs Discrete Random Vs Continous Random

1. Discrete Variable
   1. Random variable from a finite set (Pick random number from list)
2. Continuous Variable
   1. Can take ANY value in defined range eg. [0,1] but any float between that range
   2. Not contable! Infinite possb
3. Random Variable.
   1. assigns a numerical value to each outcome of a random process
   2. Always has some range

mathStackEx

Theroms (required to know!!) {Go over These Later (ask what Abt Them U Shuld know)}

todoStudy

Expected Value: (4.1,2,3)

Know: Sum of probility MUST be 1, a sum of constant must still be some constant

Rule 1:

For constant c

$$\begin{array}{r}\boxed{E(c)=c}\\ \text{proof: }\\ E(x)=\sum _{allx}xp(x)=E(c)=\dots \end{array}$$

Rule 2

$$\begin{array}{r}\boxed{E(cx)=cE(x)}\\ \text{Proof:}\\ E(cx)=\sum _{allx}cxp(x)\end{array}$$

c is some constant wrt. x, hence can bring out infront of sum. But we already have a defition for this once we bring out the c, so its $cE(x)$ proved!

Rule 3

$$\boxed{E[g_0(x)+\cdots +g_i(x)]=\sum E(g_i(x))}$$

Vairence

Defined by:

$$\begin{array}{r}\boxed{{\sigma }^2=E(x^2)-{\mu }^2}\\ \text{Proof: }\\ \text{By defition, σ2=E((x−μ)2) so expand+simplify: }\\ E(x^2-2x\mu +M^2)=\boxed{E(x^2)-2\mu E(x)+{\mu }^2}\end{array}$$

We can expand inside:

$${\sigma }^2=E(x^2)-2{\mu }^2+{\mu }^2=\boxed{E(x^2)-{\mu }^2}$$

Binomial Probaility Distro

Formula is like Combinations (Binomial Probability Distribution) Formally, in r this is described as:

dbinom(3, size = 10, prob = 0.5)

Probability of getting 3 h in 10 coin flips, whr e/a flip probability 0.5 heads

Bernouli (binomial) Probability Distribution Coin Flip Example: (will be on midterm!!)

Consider coin flip, H/T, then |sample_space|=2 (H/T)

$$\begin{array}{c}x\sim Bern(p)\\ x=\{1,0\}\end{array}$$

Then

$$\begin{array}{r}P(x=1)=P\\ P(x=0)=q=1-P\end{array}$$

Lets toss the coin n times, then count # of heads:

$$x=\text{no. heads in n flips}$$

For example

$$P(x=3)n=10p=0.4$$

Moment Generating Functions: {Math}

Defined as

$$\begin{array}{c}x\sim Bern(p)\\ \text{MGF for above: }{M}_x(t)=q_{t}p{e}^t\\ \text{Proof:}\\ \frac{dM}{dt}=\frac{d}{dt}(q+p{e}^t)=\\ \boxed{p{e}^t{|}_0=p}\end{array}$$

Use MGF derative formula (below MGF Therom)

We have 3 main defitions, all focus on the following:

$$\begin{array}{r}{\mu }_k^{\prime }=E(Y^k)\\ k=1,2,\dots \end{array}$$

MGF Therom: (ON EXAM!) See ex4.21/22

(Makes easier, instead of using defition of expected value, we can prove like so instead)

$$M_k^{\prime }=\frac{d^k{M}_x(t)}{d{A}^k}{|}_{t=0}$$

Lets have the MGF of ${\sigma }^2$

$$\begin{array}{c}{o}^2=E((x-\mu {)}^2)\\ E(x^2)-{\mu }^2\\ M_2^{\prime }-(M_1{)}^2\\ \text{Lets derive both now:}\\ E(x^2)={\mu }_2^{\prime }=\frac{d{\mu }^2}{dt}{|}_{t=0}\end{array}$$

Example: MGF (end of ch 4)

q+p =1

$$\begin{array}{c}x\sim Bin(n,p)\\ M_x(t)=(q+p{e}^t{)}^n\\ \text{Start by finding: }\frac{dM}{dt}\\ =\frac{d}{dt}(u^n)=\\ \frac{d{u}^n}{du}\cdot \frac{du}{dt}\text{<- chain rule}\\ n{u}^{n-1}p{e}^t=\boxed{n(p+p{e}^t{)}^{n-1}p{e}^t{|}_1}\end{array}$$

Overbooking problem (project 1)

Stats_overbooking_quiz TODO

see here

• Flight, n seats, y tickets,
• Find: # tickets should sell?
• $\gamma$ will always be a given var

Given some

$$\begin{array}{c}f(n)=0\end{array}$$ $$\begin{array}{c}y=\{1,0\}\\ \text{for: 1=show + ticket, 0=ticket \& no show}\\ \text{Where p=probility of showing up. We have set:}\\ S=y_1+y_2+\cdots +y_i\\ \text{then for n=no. tickets sold:}\\ S\sim Bin(n,p)?\end{array}$$

Lets start by figuring out

$$P(S\le n+1)=\gamma ?$$

Which can be denoted by the following

pbinom(N~p)

• N = no.seats (given)
• a (not given!!!)
• p=show probility (given)
• $\gamma$(given)

Code example:

• Suppose N=200 (seats on flight)
• Prob showing up $p=0.95$
• $\gamma =0.8$
• Then: we can plot with our given function

#pbinom(N+1,x,p) - 1 + g =0
 
f <- function(x, N = 200, p = 0.95, g = 0.8){
 
  obj <- pbinom(N+1, x,p) -1 +g
  abs(obj)
}
 
xx <- seq(200,220,by = 1)#f(xx) -> outputs bunch of #s
#Then to solve the problem we just need to find the smallest value for this. We can just do 
 
f(xx)
 
plot(xx, f(xx),
     pch =21,
     bg = ifelse(xx != 215, "blue", "red"),
     xlab = "Number of Tickets sold",
     ylab = "Objective function",
     main = bquote(f(x) == pbinom(N+1,x,p) - 1 + gamma))
 
which(f(xx) == min(f(xx)))#OUT 16: what index works to give us our min? Then we find our min
xx[16]
#Hence, we should sell 215 tickets given all of the input data!

Lab 5 content

Generating a Random sample

Suppose that there is a bag of 20 marbles, 12 white (“1”) and 8 black “0”. Using the sample() function create a sample of size n=5 without replacement

sample(c(rep(0,12), rep(1,8)),size=5, replace=FALSE)
#where c is some array, rep(int,repeat_x_many_times)
#size=n=#taken
#replace: should sampling be with replacement? (do I take out what I just took out of the sample space? True: False
 
# If we wanted to do a coin flip:
sample(c("H","T"),size=10,prob=c(1/2,1/2),replace=TRUE) 
sample(c(1,0),size=10,prob=c(1/2,1/2), replace=TRUE)

Binomial Experiment

Simulate a binomial experiment n=10,p=0.7, and Y=number of successes.

a <- c(100, 200, 500,1000,10000)#iteratiomns
for(i in a){
  print(  mybin(iter=i,n=18, p=0.3)  )#binomial expirment
}

Formula to code

Pois calculation

$$P(Y>4),Y\sim Pois(\lambda =2)$$

1 - ppois(q = 3, lambda = 2)

Advanced choose

$$P(Y=10),Y\sim NegBin(p=0.4,r=3)$$

choose (10 - 1, 3-1) * 0.4 ^3 * 0.6 ^ (10-3)

Advance pbinom

$$P(Y\le 8),Y\sim Bin(n=15,p=0.4)$$

pbinom(q = 8, size = 15, prob = 0.4)

Stuff on midterm

• Bernouli
• Testing problem??