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In this lecture, we continue our
discussion of continuous

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random variables.

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We will start by bringing
conditioning into the picture

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and discussing how the PDF of
a continuous random variable

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changes when we are told that a
certain event has occurred.

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We will take the occasion to
develop counterparts of some

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of the tools that we developed
in the discrete case such as

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the total probability and total
expectation theorems.

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In fact, we will push the
analogy even further.

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In the discrete case, we looked
at the geometric PMF in

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some detail and recognized an
important memorylessness

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property that it possesses.

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In the continuous case, there is
an entirely analogous story

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that we will follow, this time
involving the exponential

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distribution which
has a similar

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memorylessness property.

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We will then move to a second
theme which is how to describe

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the joint distribution of
multiple random variables.

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We did this in the
discrete case by

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introducing joint PMFs.

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In the continuous case, we
can do the same using

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appropriately defined
joint PDFs and by

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replacing sums by integrals.

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As usual, we will illustrate the
various concepts through

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some simple examples and also
take the opportunity to

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introduce some additional
concepts such as mixed random

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variables and the
joint cumulative

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distribution function.