`02:00`

STAT 1010 - Fall 2022

By the end of this lesson you should:

- Know how to use the binomial distribution, as well as its mean, and variance.
- Know how to use the Poisson distribution, as well as its mean, and variance.
- Understand differences between the binomial and Poisson distributions, and when the two could be applied.

flipping a coin

insured or not insured

luggage lost or not lost

\[\begin{equation} B =\begin{cases} 1, & \text{if the trial is a success}.\\ 0, & \text{if the trial is not a success}. \end{cases} \end{equation}\]

\(p\) is the probability of success

- \(E(B) = 0 \cdot P(B=0) + 1 \cdot P(B=1)\)
- \[\begin{aligned} E(B) &= 0 \cdot P(B=0) + 1 \cdot P(B=1)\\ &= 0 \cdot (1-p) + 1 \cdot p\\ &= p \end{aligned}\]

`02:00`

- \(Var(B) = (0 - p)^2 \cdot P(B=0) + (1 - p)^2 \cdot P(B=1)\)
- \[\begin{aligned} Var(B) &= (0 - p)^2 \cdot P(B=0) + (1 - p)^2 \cdot P(B=1)\\ &= p^2 \cdot (1-p) + (1 - p)^2 \cdot p\\ &= p^2 - p^3 + p(1 - 2p + p^2) \\ &= p^2 - p^3 + p - 2p^2 + p^3\\ &= p(1-p) \end{aligned}\]

`02:00`

This happens so often that we have another name for this distribution

\[Y = B_1 + B_2 + ... + B_n\]

- \(Y\) is the sum of independent and identically distributed random variables
- What is the mean and variance?

The number of successes in \(n\) trials

Binomial - Biceps - FIST

F - fixed number of trials

I - independent outcomes

S - probability of success is constant

T - two outcomes

\[E(Y) = E(B_1 + B_2 + ... + B_n)\]

- \[\begin{aligned} E(Y) &= E(B_1) + E(B_2) + ... + E(B_n) \\ &= p + p + ... + p \\ &= np \end{aligned}\]

`02:00`

\[Var(Y) = Var(B_1 + B_2 + ... + B_n)\]

- \[\begin{aligned} Var(Y) &= Var(B_1) + Var(B_2) + ... + Var(B_n) \\ &= p(1-p) + p(1-p) + ... + p(1-p) \\ &= np(1-p) \end{aligned}\]

`02:00`

In a class of 85 students how many groups of 4 are possible?

\(85 \cdot 84 \cdot 83 \cdot 82\)

\(48,594,840\) groups

If we don’t care about order

\(\frac{85 \cdot 84 \cdot 83 \cdot 82}{4 \cdot 3 \cdot 2 \cdot 1}\)

\(2,024,785\) groups

*Combination*written as \({}_{n}C_{k}\) here \(n = 85\), \(k = 4\) or \(\binom{85}{4}\)

If \(Y \sim Bin(n, p)\) where

- $n = $ number of trials
- $p = $ probability of success
- $y = $ number of successes in \(n\) Bernoulli trials

then

\[P(Y = y) = \binom{n}{y}p^y(1-p)^{n-y}\]

- Let \(Y \sim Bin(n = 5, p = 0.2)\) find the \(E(Y)\) and \(Var(Y)\)

- \(E(Y) = np = 1\)
- \(Var(Y) = np(1-p) = 0.8\)
- What is the probability that \(y = 3\)? In
`R: dbinom(size = 5, prob = 0.2, x = 3)`

\(0.0512\)

`02:00`

10% Condition: if trials are selected at random, it is OK to ignore dependence caused by sampling from a finite population if the selected trials make up less than 10% of the population

I know about 7 cars drive by my house in an hour, what’s the probability 18 cars drive by in the next 60 mins?

\(p = \frac{7}{60}\)

\(Y \sim Bin(60, \frac{7}{60})\)

\(\binom{60}{18}p^{18}(1-p)^{42}\)

I know about 7 cars drive by my house in an hour, what’s the probability 18 cars drive by in the next 3600 secs?

\(p = \frac{7}{3600}\)

\(Y \sim Bin(3600, \frac{7}{3600})\)

\(\binom{3600}{18}p^{18}(1-p)^{3582}\)

I know about 7 cars drive by my house in an hour, what’s the probability 18 cars drive by in the next much smaller than a second? Full derivation can be found here.

\(\binom{n}{18}(\frac{7}{n})^{18}(1-\frac{7}{n})^{n-18}\)

\(\frac{n\cdot(n-1)\ldots(n-17)}{18!}\cdot \frac{7^{18}}{n^{18}}\cdot (1-\frac{7}{n})^{n} \cdot \frac{1}{(1-\frac{7}{n})^{18}}\)

\(\frac{n\cdot(n-1)\ldots(n-17)}{n^{18}} \rightarrow 1\)

\(\frac{1}{(1-\frac{7}{n})^{18}} \rightarrow 1\)

\(\frac{7^{18}}{18!}e^{-7}\)

If \(X \sim Pois(\lambda)\) then \(P(X=x) = \frac{\lambda^{x}}{x!}e^{-\lambda}\)

- Poisson - fish - Rips
- R - randomly through space or time
- I - indepedent
- P - proportional to interval size
- S - singly - no multiple occurences in space or time

\(\lambda\)

\(\lambda\)

- Let \(Y \sim Pois(\lambda = 5)\) find the \(E(Y)\) and \(Var(Y)\)

- \(E(Y) = 5\)
- \(Var(Y) = 5\)
- What is the probability that \(y = 3\)? In
`R: dpois(x = 3, lambda = 5)`

\(0.1403739\)

`02:00`