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Flashcards in Standard distributions Deck (25):
1

What is the random variable in a Bernoulli distribution?
What are the parameters?

A variable that can take 1 of 2 values - 1 or 0. The only parameter is "p", which represents P(X = 1).

2

What is the random variable in a binomial distribution?
What are the parameters?

The number of successes after n Bernoulli trials.

The parameters are "p" and "n". "p" represents P(X = 1) for any given trial 1 to n, n being the number of trials.

3

What is the random variable in a geometric distribution?
What are the parameters?

The number of successful Bernoulli trials that occur until the first failure or the number of failures that occur until the first success.

The parameter is "p", the probability of a success.

4

What is the random variable in a Poisson distribution?
What are the parameters?

The number of times an event occurs in some finite period of time or space.

The only parameter is λ which is the rate parameter. It can be understand as "the expected number of times the event will occur"

5

What is the random variable in a negative binomial distribution?
What are the parameters?

The number of independent Bernoulli trials we must carry out until we see k success. The parameters are "k" and "p". "p" is the probability of a success

6

What is the random variable in a hypergeometric distribution?
What are the parameters?

If we have a population of N objects, precisely k of which have a special property and we take a random sample of n objects, then HypGeom(N, k, n) is the number of objects in our sample that has the special property.

The parameters are "N", "k" and "n".

7

What is the random variable in an exponential distribution?

What are the parameters?

The "time"(any continuous unit of measurement) until an event occurs.

The rate parameter λ

8

What is the random variable in a uniform distribution?

What are the parameters?

A random variable that has the same probability of landing anywhere in an interval between "a" and "b"

The parameters are "a" and "b" which represent the endpoints of the interval.

9

Expectation and variance of the Bernoulli distribution

E(X) = p, var(X) = p(1 - p)

10

Expectation and variance of the binomial distribution

E(X) = np, var(X) = np(1 - p)

11

Expectation and variance of the geometric distribution

E(X)= 1/p var(X) =1-p/p^2 for k = {1,2,3...} (Number of successes before a failure)

E(X) = (1-p)/p var(X) = (1-p)/p^2 for k= {0,1,2....} Number of failures before a success)

12

Expectation and variance of the Poisson distribution

E(X) = var(X) = λ

13

Expectation and variance of the negative-binomial distribution

E(X) = pk/(1-p) var(X) = pr/(1-p)^2

14

Expectation and variance of the hypergeometric distribution

E(X) = n(N/K) var(X) = very long...

15

Two important properties of the normal distribution

1) If A~N(μ,σ^2) and a,b are real numbers, then aX + b ~ N(aμ+ b, (a^2)σ^2)

2)If X1,...Xn are INDEPENDENT normal random variables Xi ~ N(μi, σi^2) ΣXi ~ ΣN(μi.σi)

16

What is a normalising constant?

A constant that makes certain the distribution function 's integral is equal to 1. i.e 1/(root(pi)) for the normal distribution.

17

What is Γ(1)?

1

18

if α > 1 then what is Γ(α)?

Γ(α) = (α- 1)Γ(α- 1)

19

if n is a natural number then what is Γ(n)?

(n - 1)!

20

what is Γ(1/2)?

root pi

21

Prove that the gamma function's p.d.f is a valid p.d.f

1) each term is clearly greater than 0 for x > 0

2)use lemma 2,3 to integrate the expression so that it is equal to the normalising constant.

22

How can Lemma 2.3 be varied?

integral from 0 to infinity of x^(α - 1)e-(βx) = (Γ(α))/β^x

x^(α)e-(βx) = (Γ(α + 1))/β^x

x^(α + 1)e-(βx) = (Γ(α + 2))/β^x

x^(α + 2)e-(βx) = (Γ(α + 3))/β^x

and so on

23

What is X ~ Ga(1,β) equal to?

Exp(β) as this is equal to β^1/Γ(1)

24

If X1 ~ Ga(α1, β), X2 ~ Ga (α2 , β) then...

X1 + X2 ~ Ga(α1 + α2, β)

25

When is a gamma distribution a chi-squared distribution?

When α=n/2 and β= 1/2