ST
321 Expectation and Variance
Estimating E(X):
In one of the previous classes we discussed that one
way to estimate the expectation E(X)
of a random variable X is to first
generate a large number of X’s and
then estimate E(X) by their mean. The
following R/Splus command can be used to estimate E(X) if X is an exponential random variable with rate 0.1:
> mean(rexp(10000, 0.1))
Run the above R/Splus
command and compare the result to the true E(X)=10.
Expectation of a function
of a random variable E{g(x)}:
The expected value of a
function g of a discrete random
variable X with probability density
function p(x) is the sum of g(x)*p(x) over all possible values of x, provided this sum converges
absolutely. Similarly, the expected
value of a function g of a continuous
random variable X with probability
density function f(x) is the integral
of g(x)*f(x) over all possible values
of x, provided this integral exists.
In either case, the expected value can be approximated by the sample mean of n independent simulated values of g(X). This result is a version of the
Law of Large Numbers.
Task 1.
(a).
Estimate the expectation of 3+2*X
where X is exponential with rate
2. Does E(3+2*X)=3+2*E(X)?
(b). Estimate the expected
value of |Z| where Z is standard normal. Does E(|Z|)=|E(Z)|?
(c). Estimate the expected
value of Y^2 where Y is Poisson(3).
Does E(Y^2)={E(Y)}^2?
(d). Estimate the expected
value of 1/Z, where Z is standard normal.
Variance of a random
variable X, Var(X)=E{X-E(X)}^2:
One function of a random
variable of particular interest is g(X)={X-E(X)}^2. The
expectation of g(X) is called the
variance of X, often denoted Var(X). If E(X) is known, the method in Task 1 can be used directly to
approximate Var(X).
That is, compute the mean of n
independent simulated values of g(X)={X-E(X)}^2. If E(X) is not known, replace E(X) by the sample mean and use n-1 as the divisor instead of n.
The resulting quantity is the sample variance, s^2. In R/Splus, the sample variance can be computed by the command var(x). The following
commands can be used to estimate Var(X) when X is exp(3)
with rate 3.
>
x<-rexp(10000,3)
>
sum((x-mean(x))^2)/(10000-1) #or you can do...
>
var(x)
Try the above commands.
Task 2.
(a).
Let x<-rt(10000,df=2), a
simulated t-distribution with 2
degrees of freedom. What is the mean of X?
What is the variance of X?
(b). Use s^2 to approximate the variance of a Poisson for lambda=1,2,…,99. What is the
relationship between the variance and the mean? To study the relationship, you
may want to use the following function to plot your approximated variances
against the approximated means.
task2b<-function(N=10000) {
meanarray<-seq(0,0,length=99)
vararray<-seq(0,0,length=99)
for (i in 1:99) {
x<-rpois(N, i)
meanarray[i]<-mean(x)
vararray[i]<-var(x)
}
plot(meanarray, vararray)
}
(c). Approximate the variance
of a binomial with 10 trials for p=1/100, 2/100,…,99/100.
(d). Approximate the variance
of an exponential with rate lambda. What
is the relationship between lambda and the variance?
(e). What
is the variance of 3+2*X where X is exponential with rate 2? In general, what is the variance of a+b*X where a and b are constants?