1. Conditional Probability on a joint discrete distribution
Given the joint pmf of X and Y, we want to find
and
These are the base for defining conditional distributions…
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2. Definition
For X, Y discrete random variables with joint pmf pX,Y(x,y) and marginal mass
function pX(x) and pY(y). If x is a number such that pX(x) > 0, then the
conditional pmf of Y given X = x is
Is this a valid pmf?
Similarly, the conditional pmf of X given Y = y is
Note, from the above conditional pmf we get
Summing both sides over all possible values of Y we get
This is an extremely useful application of the law of total probability.
Note: If X, Y are independent random variables then PX|Y(x|y) = PX(x).
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3. Example
Suppose we roll a fair die; whatever number comes up we toss a coin that many times. What is the distribution of the number of heads?
Let X = number of heads, Y = number on die.
We know that
Want to find pX(x).
The conditional probability function of X given Y = y is given by
for x = 0, 1, …, y.
By the Law of Total Probability we have
Possible values of x: 0,1,2,…,6.
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4. Conditional densities
If X, Y jointly distributed continuous random variables, the conditional density function of Y | X is defined to be
if fX(x) > 0 and 0 otherwise.
If X, Y are independent then
Also,
Integrating both sides over x we get
This is a useful application of the law of total probability for the continuous case.
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5. Example Consider the joint density
Find the conditional density of X given Y and the conditional density of Y given X.
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6. Conditional Expectation
For X, Y discrete random variables, the conditional expectation of Y given
X = x is
and the conditional variance of Y given X = x is
where these are defined only if the sums converges absolutely.
In general,
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7. For X, Y continuous random variables, the conditional expectation of
Y given X = x is
and the conditional variance of Y given X = x is
In general,
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8. Example
Suppose X, Y are continuous random variables with joint density function
Find E(X | Y = 2).
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9. More on Conditional Expectation
Assume that E(Y | X = x) exists for every x in the range of X. Then,
E(Y | X ) is a random variable. The expectation of this random variable is
E [E(Y | X )]
Theorem
E [E(Y | X )] = E(Y)
This is called the “Law of Total Expectation”.
Proof:
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10. Example
Suppose we roll a fair die; whatever number comes up we toss a coin that many times. What is the expected number of heads?
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11. Theorem For random variables X, Y V(Y) = V [E(Y|X)] + E[V(Y|X)] Proof:
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12. Example Let X ~ Geometric(p).
Given X = x, let Y have conditionally the Binomial(x, p) distribution.
Scenario: doing Bernoulli trails with success probability p until 1st success so X : number of trails. Then do x more trails and count the number of success which is Y.
Find, E(Y), V(Y).
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