12 - IID, Bernoulli, Poisson

Question 1

iid

Answer 1

sequence of random variables observed are independent of one another, and they are all realizations from the same identical underlying distribution

Ex. a sequence of coin tosses with the same coin, a sequence of rolls with the same die

Question 2

If the sequence of iid random variables is:

{X₁, X₂…X_n},

E(X₁ + X₂ + … + Xn) =

Var(X₁ + X₂ + … + X_n) =

sd(X₁ + X₂ + … + X_n) =

Answer 2

E = nµ_x

Var = nσ²_x

sd = sqr(n)*σ_x

Question 3

The SD of the sum of the iid random variables is ___ times the standard deviation of one of them.

(the multiplier is___, not ___)

Answer 3

sqr(n)

(the multiplier is sqr(n), not n)

Question 4

Sample Mean

Answer 4

Xbar

the sum multiplied by the constant 1/n

arithmetic average of sample values

Question 5

E(Xbar)

Var(Xbar)

sd(Xbar)

Answer 5

Sample Mean (Xbar)

E(Xbar) = µ_x

Var(Xbar) = σ²_x/n

sd(Xbar) = σ_x/n

The more observations that go into the sample mean (n increases), the less variable the sample mean becomes

Question 6

Bernoulli random variable

Answer 6

any random variable with a dichotomous outcome

takes on one of 2 values (either a 0 or a 1)

Ex. buy/don’t buy

live/die

market goes up/market goes down

Question 7

E(B)

Var(B)

sd(B)

Answer 7

Bernoulli random variable

E(B) = p

Var(B) = p(1-p)

sd(B) = sqr[p(1-p)]

*p is the probability that P(B=1)*

Question 8

Binomial random variable

Answer 8

of successes in n iid Bernoulli trials

Y = B₁ + B₂+…+ B₁₀₀

the sum of n, iid Bernoulli trials with success probability p

Y ~ Bi(n, p)

Question 9

E(Y)

Var(Y)

sd(Y)

Answer 9

E(Y) = np

Var(Y) = np(1-p)

sd(Y) = sqr[np(1-p)]

Question 10

Bbar

Answer 10

sample mean of the Bernoulli

Bbar = Y/n

proportion of successes → in special case of 0/1 outcome variables, the mean has an interpretation as the proportion of 1’s

Question 11

E(Bbar)

Var(Bbar)

sdBbar)

Answer 11

E(Bbar) = p

Var(Bbar) = [p(1-p)]/n

sd(Bbar) = sqr[(p(1-p))/n]

Question 12

interpret Bi(10, 0.5)

Answer 12

if 10 trials with 0.5 probability for each event

Question 13

Probability distribution for Binomial random variables

Answer 13

• binomial random variable can only take on whole number values ranging from 0 to n
• prob distribution is centered around its mean
• prob distribution is very close to bell-shaped

Question 14

Ex. if p=0.10, what is P(Y=100)?

Answer 14

0.1¹⁰⁰

Question 15

Binomial probability function

Answer 15

p(Y=y) = _nC_yp^y(1-p)^n-y

where y = 0, 1,…, n

Ex. probability of getting exactly 2 heads when you toss a fair coin 5 times.

Y ~ Bi(5, 0.5)

p(Y=2) = ₅C₂ * .5² * (1-.5)^5-2 = .3125

Question 16

Poisson random variables

Answer 16

count the number of events that occur in some time interval or given area

has lambda = Poisson parameter = expected rate that events occur at, over a given time period

*no “n” like in the Binomial number of trials*

Ex. number of hurricanes in a given yr

number of chocolate chips in a chocolate chip cookie

Question 17

Poisson probability distribution

Answer 17

P(Y=y) = e^-lambda(lamda^y)/(y!), y = 0, 1, 2…

as lambda increases, Poisson probability distribution becomes more normal (lamda=25)

Question 18

E(Y)
Var(Y)

sd(Y)

for Poisson

Answer 18

E(Y) = lamda

Var(Y) = lamda

sd(Y) = sqr(lamda)