Let X ∼ Pois(4), Y ∼ Pois(12), U ∼ Pois(3) be independent random variables. a) What...

Let X ∼ Pois(4), Y ∼ Pois(12), U ∼ Pois(3) be independent random variables.

a) What is the exact sampling distribution of W = X + Y + U?
b) Use R to simulate the sampling distribution of W and plot your results. Check that the simulated mean and standard error are close to the theoretical mean and standard error.
c) Use the simulated sampling distribution to estimate P(W ≤ 14) and then check your estimate with an exact calculation.

Expert Solution

please have a look at the R code

rx=rpois(500,4) #We have created 500 random observations from poisson distribution
ry=rpois(500,12)
ru=rpois(500,3)
rw=rx+ry+ru
plot(c(1:500),rw,type='p')
mean_w=mean(rw)
mean_w
sd_w=sqrt(var(rw))
sd_w
boolean_14=rw<=14 #It's a boolean vector giving TRUE if observation is less than 14
p=(sum(boolean_19))/500 #Sum(boolean_14) gives us number of rw observations which are less than 14
p
true_p=ppois(14,19) # It gives theoretical P(W<=14)
true_p

"Now have a look at the output"

simulated mean is coming out to be theoretical mean and simulated standard devaition (4.325) is very close to actual sd ( root(19)=4.3589). The simulated probabolity (0.146) is very close to actual probability (0.1497)

milcah answered 7 hours ago

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