Distribution of Bernoulli random variable

[trojansc82]

Homework Statement

a) Let X₁, X₂, ...X_N be a collection of independent Bernoulli random variables. What is the distribution of Y = \sum^N_{i = 1} X_i


b) Show E(Y) = np

Homework Equations

Bernoulli equations f(x) = p^x(1-p)^1-x

The Attempt at a Solution

a)X₁ + X₂ + ... + X_N = p

b) Not sure. I'm assuming the expectation would mean when each probability is multiplied by "x", the x's go from 0 to n, meaning they represent the n.

 

[LCKurtz]

Homework Statement

a) Let X₁, X₂, ...X_N be a collection of independent Bernoulli random variables. What is the distribution of Y = \sum^N_{i = 1} X_ib) Show E(Y) = np

Homework Equations

Bernoulli equations f(x) = p^x(1-p)^1-x

The Attempt at a Solution

a)X₁ + X₂ + ... + X_N = p

b) Not sure. I'm assuming the expectation would mean when each probability is multiplied by "x", the x's go from 0 to n, meaning they represent the n.

Have you stated the question carefully and correctly? What is p? Are the independent random variables of the same distribution? Have you studied binomial distributions yet?

 

[trojansc82]

Have you stated the question carefully and correctly? What is p? Are the independent random variables of the same distribution? Have you studied binomial distributions yet?

Yeah, I have studied the binomial. I'm going back and looking at old tests before my final, so I'm trying to remember the way the solution worked.

I was easily able to derive the mean and variance from a Bernoulli distribution, but I'm having trouble with these two problems.

Also, p is the expected outcome. I'm assuming you're just deriving the mean of the Binomial distribution here, but I'm not sure how that is done.

 

[LCKurtz]

The fact that the sum of n IID Bernoulli RV's gives a binomial RV is standard stuff. It's probably worked out in your text and is all over the internet. One proof is here:
http://en.wikipedia.org/wiki/Binomial_distribution