What is the general formula for the area of the shape that you see
2 answers:
You might be interested in
Answer:
Step-by-step explanation:
Let's start defining the random variables for this exercise :
'' The proportion of the particulates that are removed by the first pass ''
'' The proportion of what remains after the first pass that is removed by the second pass ''
'' The proportion of the original particulates that remain in the sample after two passes ''
We know the relation between the random variables :
We also assume that and
are independent random variables with common pdf.
The probability density function for both variables is for
and
otherwise.
The first step to solve this exercise is to find the expected value for and
.
Because the variables have the same pdf we write :
Using the pdf to calculate the expected value we write :
Where ∞ and
- ∞ (because we integrate in the whole range of the random variable). In this case, we will integrate between
and
⇒
Using the pdf we calculate the expected value :
⇒
Now we need to use some expected value properties in the expression of ⇒
⇒
Applying the expected value properties (linearity and expected value of a constant) ⇒
Using that and
have the same expected value
and given that
and
are independent random variables we can write
⇒
⇒
Using the value of calculated :
We find that the expected value of the variable is