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Polya urn simulation via R

I need help in two probability problems using R program. Thanks so much!
1. Simulation of an expectation. Design a simulation exercise in R to approximate the value of E ( X^X)
when X has a U [0,1] distribution. Plot the value of the approximation as the sample size goes from 1 to
10,000. Use the Notebook feature in RStudio to create an HTML file for your answer.

2. Simulation of a Polya Urn. Put a = 10 white balls and b = 20 blue balls in an urn. Then, randomly draw
a ball from the urn, and replace it along with another ball of the same color, repeat.
a) Write an R program to simulate Rn, the proportion of white balls in the urn after n draws from the urn,
for any integer n.
b) Plot the histogram of Rn for n = 1,000.
c) Simulate Rn 10,000 times.
d) Overlay the histogram with a line graph of the appropriate limiting distribution. Use the Notebook
feature in RStudio to create an HTML file for your answer.

Solution Preview

1. The code is given below. We simply sample from a uniform with the given sample size to estimate E(X^X). The last line should give you the plot you need.

means <- rep(NA, 10000)
for (sample_size in 1:10000) {
x <- runif(sample_size)
means[sample_size] <- mean(x^x)
}
plot(means)

2. Below we have the R ...

Solution Summary

The solution provides the full code for both questions, with an explanation of what the key components mean.

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