Characteristic function of Sum of Random Variables

[cutesteph]

Homework Statement

Let X,W,Y be iid with a common geometric density f_x(x)= p(1-p)^x for x nonnegative integer
and p is in the interval (0,1)

What is the characteristic function of A= X-2W+3Y ?

Determine the family of the conditional distribution of X given X+W?

Homework Equations

the characteristic function of the geomtric series is p/[1-(1-p)exp(it)]

The Attempt at a Solution

The characteristic function of a sum of random variables is the product of the individual characteristic functions.

So I need to find the characteristic function of X , -2W and 3Y and multiply them together?

 

[haruspex]

Sounds right to me. (It requires them to be independent, which they are.)

 

[cutesteph]

This may sound like a stupid question, but how do I get the density of a discrete random variable when multiplied by a constant? Do I simply subsistute the value in for the variable. Like if I has a binomial random variable X~ Bin (n,p) (I choose n) p^i (1-p)^n-1 for i= 0,1,2,...,n how would I find the density of 5X?

 

[haruspex]

This may sound like a stupid question, but how do I get the density of a discrete random variable when multiplied by a constant? Do I simply subsistute the value in for the variable. Like if I has a binomial random variable X~ Bin (n,p) (I choose n) p^i (1-p)^n-1 for i= 0,1,2,...,n how would I find the density of 5X?

Maybe, but I wouldn't assume that. Start with the def of characteristic function and try to work it out from first principles. Shouldn't be hard.

 

[Ray Vickson]

This may sound like a stupid question, but how do I get the density of a discrete random variable when multiplied by a constant? Do I simply subsistute the value in for the variable. Like if I has a binomial random variable X~ Bin (n,p) (I choose n) p^i (1-p)^n-1 for i= 0,1,2,...,n how would I find the density of 5X?

Of course, discrete random variables do not have densities, but the do have probability mass functions and (cumulative) distribution functions. If p(k), k in K, is a discrete pmf (so that P{X = k} = p(k)) then for Y = 5X, the pmf is P{Y = j} = P{5X=j} = P{X=j/5} = p(j/5) (but only for values of j such that j/5 is in K).

RGV

 

[cutesteph]

So for example if I wanted 2X where X ~Bin(5,1/2) ie (5 choose x)(1/2)^x (1/2)^5-x for x =0,1,2,3,4,5. Then 2X would be Bin(2,1/2) (2 choose x)(1/2)^ (1/2)^2-x for x=0,1,2?

How would I find a joint probably mass function of two binomials? Let's say was was Y= 2X and the other was Z=3x and I want the joint pmf of (Y,Z)?

 

[Ray Vickson]

So for example if I wanted 2X where X ~Bin(5,1/2) ie (5 choose x)(1/2)^x (1/2)^5-x for x =0,1,2,3,4,5. Then 2X would be Bin(2,1/2) (2 choose x)(1/2)^ (1/2)^2-x for x=0,1,2?

How would I find a joint probably mass function of two binomials? Let's say was was Y= X+2X and the other was Z=-2X+3X and I want the joint pmf of (Y,Z)?

No. If X ~ Bin(5,1/2) and Y = 2X, we have P{Y = j} = P{X = j/2} = C(5,j/2)/2^5 for j = 0,2,4,6,8,10, and P{Y = j} = 0 for all other j.

RGV

 

[cutesteph]

Also would the characteristic function of A in my orginal question would be phi(t) = [p/(1-(1-p)exp(it))] [p(1-p)/(1-p-exp(it))] [p/(1-exp(it)(1-p)^3] .

 

[cutesteph]

No. If X ~ Bin(5,1/2) and Y = 2X, we have P{Y = j} = P{X = j/2} = C(5,j/2)/2^5 for j = 0,2,4,6,8,10, and P{Y = j} = 0 for all other j.

RGV

So what happens when I had binomials with different indexes? like X+Y in the quoted example?

 

[Ray Vickson]

So what happens when I had binomials with different indexes? like X+Y in the quoted example?

You know the basic formulas for probabilities of a sum of independent random variables, and you know the probability mass functions of the individual random variables. Just put it altogether yourself.

RGV

 

[cutesteph]

You know the basic formulas for probabilities of a sum of independent random variables, and you know the probability mass functions of the individual random variables. Just put it altogether yourself.

RGV

Got it. Thanks.



Determine the family of the conditional distribution of X given X+W?

X+W is a negative binomial(2,p) since the product of the moment generating function X and W shares the same moment generating function with a binomial (2,P) , which we can do since they are independent and we know that moment generating functions are unique.

We know the negative binomial counts the number of failures proceeding the 2 sucess (in this case) in a sequence of bernoulli trials.

The geometric distribution counts the number of bernoulli trials to get one sucess.


What does it mean by family of the conditional distribution?

 

[haruspex]

What does it mean by family of the conditional distribution?

I would assume it's asking whether it's binomial or whatever.
It would probably be useful to think about the joint distribution of X and X+W.