Unit 7: Random Variables and their PDFs

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 Milestone

This milestone develops insight into random variables. Thanks to Chris Olsen for creating these.

1. Warm-up

Suppose that random variables D= the number of pips after rolling one fair die and C = number of heads after flipping one fair coin.

a) Calculate the mean and the standard deviation for D and for C.

b) Now define a new random variable D + C. Calculate its mean and standard deviation by finding the pdf for D + C.

c) Now compare your results to the mean and SDs you get for using the "rules" for adding random variables. How do these compare?

2. Racing on the Island of Nog

On the Island of Nog there are three species: horses, three-toed sloths, and humans. We classify horses as one-toed animals. We randomly select a creature from the island. If we define the random variable T = number of toes on one foot, then the probability distribution of T is
t 1 3 5
P(T=t) .4 .1 .5

a) From the table, calculate the mean, variance, and SD of T.

The annual Nog Spring Picnic features a two-legged race, where creatures are paired together in the following manner: The left hind leg of a randomly selected creature is bound to the right hind leg of another randomly selected creature. We define a new variable S to be the sum of toes for the bound pair, X = number of toes of the first creature and Y = number of toes of the second creature.

b) Now construct a table for S. there are nine ways to make teams.

c) What is the probability that a team will have more than five toes?

d) What is the probability that a team will have more than 8 toes, given that one member of the team is human?

e) Find the mean and variance for S.

f) Because of the differences between the species, the committee suggests that some teams be given a "head start", depending on the number of toes of the bound legs. The number of meters of head start is determined by the formula: H = 3 + 7S. Calculate the mean and the variance for H.

Write up your solutions and name them ms7yourlastname.pdf and drop it in the Digital Dropbox.