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Question: r code program 1 suppose we want to simulate an...

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1. Suppose we want to simulate an experiment that can take outcomes 1; : : : ; n with probabilies p1; : : : ; pn. To be specic, suppose the R-vector p=c(.1,.2,.3,.35, .02, .03) gives the desired probabilities. Write R code that produces a number from 1 to 6 with the given probabilities, without using if statements. I recommend using the R command cumsum to do this, though there many possible approaches.

2. Suppose we are interested in P(A) for some event A. While the n^(1/2) rule

is simple, a better 95% confidence interval is given by

p +- 1.96((p(1-p)/n)^(1/2))

where ^p is still the proportion of times A occurs in our sample. This

confidence is smaller, which is better, but still contains the true probability

about 95% of the time.

Suppose our event of interest has P(A) = 1/10.

(a) Simulate 300 trials of the experiment and print out the resulting

confidence interval using the formula above.

(b) In your simulation above, for each trial, 1,. . . , 300, plot the upper

and lower endponts of the confidence interval. That is, if n is the

number of trials done so far, and un and ln are the upper and lower

limits of the confidence interval based on n trials, plot the points

(x; y ) = (n; Un ) and (x; y ) = (n; Ln ) for n = 1; : : : ; 300.

(c) Repeat the experiment above 1000 times and compute the fraction of

the time the confidence interval contains the true probability, 1/10.

This should happen about 95% of the time.

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