Question

R programming question

a) Do in R OR by hand

b) Do using R only. Don’t forget confidence interval.

. The commute time between LPC and Loop campus is X+ Y, where X is the time in minutes to wait for train and Y is the time in minutes for the train travel. Suppose that X has a zero-inflated exponential distribution, i.e., the cumulative probability distribution is: 0, 0.4 1 0.6e-/5 -oo x<0, Fx(x) 0 r oo. Suppose that Y is a normal random variable with mean 20 minutes and variance 25 minutes squared (a) (5 points) Compute the true mean time it takes to get to LPC, a = E(X+ Y) (b) (10 points) Perform a simple Monte Carlo simulation to estimate u using 1000 samples. Compute and a 99% confidence interval for

Answer 1

(a) Construct two dummy independent random variables U and V such that

P(U=0)=.4 and P(U=1)=.6 and V~Exponential(mean=5). Then for x>0,

P(UV<=x)=P(UV<=x,U=0)+P(UV<=x,U=1)=P(U=0)+P(V<=x,U=1)

=.4+(1-exp(-x/5))*.6

=1-.6 exp(-x/5).

And, for x=0, P(UV<=0)=P(U=0)=.4. Hence X has the same distribution as the product UV.

Then E(X+Y)=E(UV)+E(Y)=E(U)E(V)+E(Y)=.6*5+20=23 (as E(U)=.6 and E(Y)=20).

(b)  ## For the 99% CI, we have constructed a normal approximation based CI.

R Program

nsim=1000
s=numeric(nsim)
for(i in 1:nsim)
{
u=rbinom(1,1,.6)
v=rexp(1,rate=1/5)
x=u*v
y=rnorm(1,20,5)
s[i]=x+y
}
mu.hat=mean(s)
lcl=mean(s)+sd(s)*qnorm(.005)/sqrt(nsim)
ucl=mean(s)-sd(s)*qnorm(.005)/sqrt(nsim)
# Monte Carlo estimate
mu.hat
#99% Confidence interval of mu
lcl
ucl

R Output

> # Monte Carlo estimate
> mu.hat
[1] 22.99667
> #99% Confidence interval of mu
> lcl
[1] 22.82015
> ucl
[1] 23.17318

For any query, comment.