Math488 flashcards midterm 2

what is the x and y in the PDF graph?

x-results of random variable

y-probabilities

using (α, β), an exponential is a Gamma with what parameters?

alpha = 1 and beta (1/mean) Γ(1,β)

The Chi Squared distribution with one degree of freedom represented as a Gamma with parameters (α, β)

Γ(1/2, 2)

using lambdas, x and alpha, what is the PDF of the gamma?

The sum of Chi-Squared distributions is what as gamma parameters? How many degrees of freedom is from alpha = n/2? What is this denoted?

Alpha is the degrees of freedom over 2. So if n =2 alpha = 1, Sn is a chi-square distribution with n degrees of freedom

Let X be a random variable. We have a Y random variable involving an equation with X and tao. When is the distribution called transformed and what is this equation?

If τ is -1, then the resulting distribution is called..

the inverse

If τ < 0 but not -1, then the distribution is called?

the distribution is called inverse-transformed or inverse-generalized

For what value of τ is there no transformation?

τ=1

Formual for a transformed random variable fY(y) = …. What are the two formulas with different inputs into that fox?

What is X and Y the same as here?

g-1(y), g(x)

Represent the above X and Y in terms of Tao alongside dx/dy (derivative of X portion with respect to y)

Now go back to our formula with fY(Y)=fX(X)*dx/dy and plug in X and dx/dy

decompose X^(1/τ) now that X = theta subscript x *W

So when intermingling between the Gamma formulas based on our parameters, what is the principle here?

We can transform our random variable to a transformed gamma distribution

What is the Weibull distribution?

The windell distribution is a special case of the transformed Gamma distribution with alpha =1. For the PDF of the weibull remember that u = x/theta

How does the exponential relate to the Weibull?

The exponential is a special case of the Weibull with τ=1

And the exponential is a special case of the transformed Gamma with what?

τ=1 and alpha =1

Definition of the linear exponential family

the Gamma is a special case of the ___

transformed Gamma with T = 1

The support of a random variable

The set of values where the PMF/PDF is non-zero

Explain the Unsum property

Suppose the number of occurrences N is a Poisson R.V with E[X] lambda. Further suppose each occurrence can be classified into one of m types each with probabilities p1 to pm independent of all other occurrences. N1 .. Nm corresponding to types 1…m respectively are mutually independent poisons with expectations p1* lambda1, p2*lambda 2, and p3 * lambda 3.

If we remove a type that accounts for 10 percent of the claims and the original expectation of all the claims was 0.72, what is the new expectation?

0.72×0.9

Expectation for trial and variance for negative binomial

r/p and r*q/p²

Two PMFs for negative binomial

Expectation for failure negative binomial

r/p-r

x! relation to gamma

x! = gamma function(x+1)