We now look at an example similar to the previous one,
in which we have again two scenarios, but in which we
have both discrete and continuous
random variables involved.
You have $1 and the opportunity
to play in the lottery.
With probability 1/2, you do nothing and you're left with
the dollar that you started with.
With probability 1/2, you decide to play the lottery.
And in that case, you get back an amount of money which is
random and uniformly distributed
between zero and two.
Is the random variable, X, discrete?
The answer is no, because it takes values on
a continuous range.
Is the random variable, X, continuous?
The answer is no, because the probability that X takes the
value of exactly one is equal to 1/2.
Even though X takes values in a continuous range, this is
not enough to make it a continuous random variable.
We defined continuous random variables to be those that can
be described by a PDF.
And you have seen it in such a case, any individual point
should have zero probability.
But this is not the case here, and so X is not continuous.
We call X a mixed random variable.
More generally, we can have a situation where the random
variable X with some probability is the same as a
particular discrete random variable, and with some other
probability it is equal to some other
continuous random variable.
Such a random variable, X, does not have a PMF because it
is not discrete.
Also, it does not have a PDF because it is not continuous.
How do we describe such a random variable?
Well, we can describe it in terms of a cumulative
distribution function.
CDFs are always well defined for all
kinds of random variables.
We have two scenarios, and so we can use the Total
Probability Theorem and write that the CDF is equal to the
probability of the first scenario, which is p, times
the probability that the random variable Y is less than
or equal to x.
This is a conditional model under the first scenario.
And with some probability, we have the second scenario.
And under that scenario, X will take a value less than
little x, if and only if our random variable Z will take a
value less than little x.
Or in CDF notation, this is p times the CDF of the random
variable Y evaluated at this particular x plus another
weighted term involving the CDF of the random variable Z.
We can also define the expected value of X in a way
that is consistent with the Total Expectation Theorem,
namely define the expected value of X to be the
probability of the first scenario, in which case X is
discrete times the expected value of the associated
discrete random variable, plus the probability of the second
scenario, under which X is continuous, times the expected
value of the associated continuous random variable.
Going back to our original example, we have two
scenarios, the scenarios that we can call A1 and A2.
Under the first scenario, we have a uniform PDF, and the
corresponding CDF is as follows.
It's flat until zero, then it rises linearly.
And then it stays flat, and the value
here is equal to one.
So the slope here is 1/2.
So the slope is equal to the corresponding PDF.
Under the second scenario, we have a discrete, actually a
constant random variable.
And so the CDF is flat at zero until this value, and at that
value we have a jump equal to one.
We then use the Total Probability Theorem, which
tells us that the CDF of the mixed random variable will be
1/2 times the CDF under the first scenario plus 1/2 times
the CDF under the second scenario.
So we take 1/2 of this plot and 1/2 of that plot
and add them up.
What we get is a function that rises now at the slope of 1/4.
Then we have a jump, and the size of that to jump is going
to be equal to 1/2.
And then it continues at a slope of 1/4 until it reaches
this value.
And after that time, it remains flat.
So this is a simple illustration that for mixed
random variables it's not too hard to obtain the
corresponding CDF even though this random variable does not
have a PDF or a PMF of its own.