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