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Magazine Chapter 5: Q40E (page 229)
A box contains ten sealed envelopes numbered\({\rm{1, \ldots ,10}}\). The first five contain no money, the next three each contains\({\rm{\$ 5}}\), and there is a \({\rm{\$ 10}}\) bill in each of the last two. A sample of size \({\rm{3}}\) is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the amounts in the selected envelopes, the statistic of interest is \({\rm{M = }}\) the maximum of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\).
a. Obtain the probability distribution of this statistic.
b. Describe how you would carry out a simulation experiment to compare the distributions of \({\rm{M}}\) for various sample sizes. How would you guess the distribution would change as \({\rm{n}}\) increases?
Short Answer
a) The probability distribution of this statics
b) As \({\rm{n}}\) increases, the last probability, \({\rm{P(M = 5)}}\), would likewise decline, although at a slower rate than \({\rm{P(M = 0)}}\).
Step by step solution
Definition
Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.
Obtaining the probability distribution of this statistic
(a):
Random variable \({\rm{M}}\) can take values\({\rm{0,5}}\), and\({\rm{10}}\). It is easiest to calculate the probability of event\({\rm{\{ M = 0\} }}\), because the only way for maximum to be \({\rm{0}}\) is if all \({\rm{3}}\) selected envelops contain no money. Therefore,
\({\rm{P(M = 0) = 0}}{\rm{.5 \times 0}}{\rm{.5 \times 0}}{\rm{.5 = 0}}{\rm{.125}}{\rm{.}}\)
In order for maximum to be \({\rm{10}}\) , there has to be at least one \({\rm{10}}\) dollar envelop in the sample. Complement of the event that there is at least one is that there is no a single envelope containing \({\rm{10}}\) dollar bill, therefore
\(\begin{aligned}\rm P(M = 10) &= 1 - P(0 cnvelopes with \$ 10 bill )\\\rm &= 1 - 0{\rm{.}}{{\rm{8}}^{\rm{3}}}\\\rm &= 0{\rm{.188}}\end{aligned}\)
where \({\rm{0}}{\rm{.8 = 0}}{\rm{.5 + 0}}{\rm{.3}}\), standing for the other two envelops.
Finally, the last probability can be calculated as
\(\begin{aligned}{\rm{P(M = 5)}}\rm &= 1 - P(M = 0) - P(M = 10)\\\rm &= 1 - 0{\rm{.125 - 0}}{\rm{.488}}\\\rm &= 0{\rm{.488}}\end{aligned}\)
The pmf of random variable \({\rm{M}}\) can be represented in a table as follows
Note: there are other ways to compute the probabilities. This one should be the easiest.
Step 3: The distribution would change as \({\rm{n}}\) increases Lengthy step
(b):
The given random variable in the exercise has a discrete distribution with pmf
There is a well-known simulation approach. The first stage would be to generate random digits from \({\rm{0}}\) to\({\rm{9}}\). (uniformly). Each number indicates a value; for example, digits \({\rm{0 to 4}}\) (total of \({\rm{5}}\) digits) represent\({\rm{0}}\), digits \({\rm{5 to 7}}\) (total of \({\rm{3}}\) digits) represent\({\rm{5}}\), and digits\({\rm{8}}\) to \({\rm{9}}\) (total of \({\rm{2}}\) digits) represent \({\rm{10}}{\rm{.}}\)
We may compare the distributions of \({\rm{M}}\) for various sample sizes by producing a sample of a certain size (number of digits chosen at random) and computing the statistic \({\rm{M}}\) from each sample.
When \({\rm{n}}\) rises, the probability varies. Probability \({\rm{P(M = 0)}}\) would decline as \({\rm{n}}\) increased, tending to zero, but probability \({\rm{P}}\left( {{\rm{M = 10}}} \right)\) would grow, heading to one. As \({\rm{n}}\) increases, the last probability, \({\rm{P(M = 5)}}\), would likewise decline, although at a slower rate than \({\rm{P(M = 0)}}\).
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