01
Function on one variable
Suppose the PDF of
is given by: Find the CDF and PDF of
.
Solution
01
(1) Start by finding the CDF of
: Let us compute
: Thus, we have that:
(2) Finally, differentiate to find the density
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02
PDF of Min and Max
Suppose
and and these variables are independent. Find: (a) The PDF of
(b) The PDF of
Solution
02
(a)
(1) First, we define the PDF’s and CDF’s of
and :
(2) Now:
since, by definition,
. By independence:
Thus, we have:
and thus,
(b)
(1) Similarly, for
, we have that . Thus, we have: by independence. Thus, we have that:
(2) Thus, the density function is given by:
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03
PDF of a Min
Let
and be independent copies of a random variable. Let
. Find the PDF of .
Solution
12
(1) Write PDFs:
PDFs of
and : Joint PDF using independence:
(2) CDF of Min:
Write
. Then:
(3) Evaluate with formula for
: For
: Therefore:
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04
Coffee and danish
Joe visits a coffee shop every morning and orders a triple espresso and a cherry danish. Let
represent the wait time for the espresso and the wait time for the danish (both in minutes). The joint PDF of and is: Suppose that Joe will not sit down until he has both his espresso and his danish. Compute the probability that he will have to wait at least 5 minutes before sitting down, that is, find
.
Solution
