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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.

(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.

(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 \({\rm{P(M = 0)}}\).