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This exercise is using two independent exponential random variables, to illustrate the conditional density function.
Here we have , , as our independent exponential random variables, which have as mean respectively:
And now we need to find the conditional density of and , which .
Solution
We can set the joint density of first, and then we can have the joint density of , which
In this step, we need to confirm that must be independent.
So we can derive the formula:
After the derivation, we can pack some variables into
And then we can discuss the following two scenarios:
Case 1:
And we will get:
Case 2:
And we can get:
Also we can have the byproduct of :
Example 3.10 (Expectation)
The Expectation of the sum of a Random Number of Random Variables
Now we calculate the expectation of compound random variables. Consider the case: we have a random number of accidents in each day, and each accident will cost some money which also use a random variable to represent.
So here we can have:
We can derive the equation:
Need to consider , the expectation of this random number, and we can get the final state: