Expectation and variance of a random number of random variables

In summary, the conversation discussed finding the expected value and variance of a random variable Z, which is the sum of independent and identically distributed random variables X1 to XN. The formula for expected value and variance were provided, along with the law of total variance. The first part of the question was solved by using the total variance formula, and the second part involved finding the probability mass function, probability generating function, and expected value of Z given a fixed value of N. The relation between the formulas used was also discussed.
  • #1
Kate2010
146
0

Homework Statement



Let X1...XN be independent and identically distributed random variables, N is a non-negative integer valued random variable. Let Z = X1 + ... + XN (assume when N=0 Z=0).
1. Find E(Z)
2. Show var(Z) = var(N)E(X1)2 + E(N)var(X1)

Homework Equations



E(Z) = EX (E(X|Z))
Law of total variance: var(Z) = EX (var(Z|X)) + VarX (E(Z|X))

The Attempt at a Solution



1. I think I have managed this, I got E(N)E(X)
2. I'm unsure how to tackle this one, I know var(Z) = E(Z2) - E(Z)2, and I know E(Z)2 but I don't know how to calculate the other, or if I should be using the equation above, and if so, how.
 
Physics news on Phys.org
  • #2
2. follows immediately from the total variance formula.

var Z=E ( var (Z|N)) + var (E (Z|N))

E ( var (Z|N))=E(N var (X1))=var(X1) E(N) -- by fixing N, Z is a sum of fixed number of Xi-s

var (E (Z|N))=var (N E(X1))=[E(X1)]^2*var(N)
 
  • #3
Thanks :)
 
  • #4
The question has a second part which I've just attempted but am also struggling with:

The number of calls received each day at an emergency centre, N, has a poisson distribution, with mean [tex]\mu[/tex]. Each call has probability p of requiring immediate police response. Let Z be the random bariable representing the number of calls involving police response.

a) What is the probability mass function of Z given N=n?
b) What is the probability generating function of Z, given that we know N=n?
c) Find E(sZ) (use the partition theorem for expectation)
d) Deduce the unconditional distribution of Z and write down var(Z).
e) How is this related to the formula we already worked out?

a) If we know N=n, can we model Z on a binomial distribution with parameters (n,p) so pZ(n) = ([tex]^{N}_{n}[/tex])pn(1-p)N-n =pn
b) The binomial p.g.f. is (q+ps)n
c) E(sZ) = [tex]\sum[/tex][tex]^{N}_{n=1}[/tex] E(sZ | N=n)P(N=n) = [tex]\sum[/tex][tex]^{N}_{n=1}[/tex] (q+ps)npn = [tex]\sum[/tex][tex]^{N}_{n=1}[/tex] (pq + p2s)n

I don't understand where I go from here if any of that is correct. I'm not sure that I should have modeled Z on a binomial r.v.
 

1. What is the difference between expectation and variance?

The expectation of a random variable is the average value that we would expect to obtain if we were to take an infinite number of samples from the distribution. It is also known as the mean. The variance, on the other hand, measures the spread or variability of the data around the mean. It tells us how much the data deviates from the expected value. In simpler terms, expectation is a measure of central tendency while variance is a measure of dispersion.

2. How is the expectation of a random number of random variables calculated?

The expectation of a random number of random variables is calculated by taking the sum of the individual expectations of each variable multiplied by their respective probabilities. In other words, it is the weighted average of the expectations of the individual variables.

3. What does a high variance of a random number of random variables indicate?

A high variance of a random number of random variables indicates that there is a wide range of possible outcomes for the given set of variables. This could suggest that the data is highly diverse and unpredictable, with a larger spread of values around the mean.

4. How does increasing the number of random variables affect the expectation and variance?

Increasing the number of random variables will generally increase the expectation as there are more variables contributing to the overall average. However, the effect on variance is less predictable. It may increase or decrease depending on the specific values of the individual variables and their probabilities.

5. Can the expectation and variance be the same for two different sets of random variables?

It is possible for the expectation and variance to be the same for two different sets of random variables. This can occur when the individual variables and their probabilities are carefully chosen to balance each other out, resulting in the same overall mean and variability. However, this is not a common occurrence and usually requires specific conditions to be met.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
694
  • Calculus and Beyond Homework Help
Replies
1
Views
874
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
421
  • Calculus and Beyond Homework Help
Replies
2
Views
923
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Back
Top