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When to use conditional pmf and conditional cdf?


Asked by Bryson Sheppard on Dec 01, 2021 FAQ



We have discussed conditional probability before, and you have already seen some problems regarding random variables and conditional probability. Here, we will discuss conditioning for random variables more in detail and introduce the conditional PMF, conditional CDF, and conditional expectation.
Keeping this in consideration,
And this leads us to this definition of conditional PMFs. The conditional PMF is defined to be the ratio of the joint PMF-- this is the probability that we have here-- by the corresponding marginal PMF. And this is the probability that we have here.
In respect to this, One way of modeling two random variables is by specifying the joint PMF. But we now have an alternative, indirect, way using the multiplication rule. We can first specify the distribution of Y and then specify the conditional PMF of X for any given value of little y.
Consequently,
Similarly, we define the conditional CDF of X given A as FX | A(x) = P(X ≤ x | A). In some problems, we have observed the value of a random variable Y, and we need to update the PMF of another random variable X whose value has not yet been observed.
Additionally,
The conditional probability-- that X takes on a specific value, given that the random variable Y takes on another specific value. And we use this notation to indicate those conditional probabilities. As usual, the subscripts indicate the situation that we're dealing with.