- #1
dirk_mec1
- 761
- 13
Homework Statement
I know that per definition [tex]E(N)= \sum P(N=k) \cdot k [/tex]. But how can I rewrite the above expectation towards the 'usual definition'?
Well... its the expectation just written in another form which I'll have to proof.statdad said:First, I've never seen the result you are trying to prove - but that doesn't prove it is written incorrectly. I know of the following result:
If the random positive random variable (so that [tex] F(0-) = 0 [/tex], then
[tex]
E(X) = \int_0^\infty \Pr(X > x) \, dx = \int_0^\infty \Pr(X \ge x) \, dx
[/tex]
although the integral may be infinite. Is this what you are discussing?
Yes you're right so we get:Second, your integrals, as written, don't make any sense - you need to integrate with respect to two variables, not just one. To point:
[tex]
\int_0^\infty \, \int_x^\infty f(y) \, dy
[/tex]
is meaningless without a [tex] dx [/tex] as well.
dirk_mec1 said:What do you propose then?
statdad said:To answer
what you're doing in the double integral is not a change of variable - if that were the case, your step could be correct.
Try going through the same steps without changing from [tex] f(y) [/tex] to [tex] f(x) [/tex] at the aforementioned point. (And remember that when you integrate from [tex] 0 [/tex] to [tex] y [/tex] w.r.t. [tex] x, f(y) [/tex] will act like a constant.
Yes, you're right it should be:statdad said:When you write
you are essentially writing
[tex]
\Pr(N \ge k) = \sum_{l=1}^n \Pr(N=l)
[/tex]
This is not correct - the sum on the right here equals 1. In short, the expression for [tex] \Pr(N \ge k)[/tex] needs to be fixed.
Once that is one, you will have a double sum: reversing the order of summation (watch the indices) will get you where you need to be.
A stochastic process is a collection of random variables that evolves over time. It is used to model systems that involve some element of randomness or uncertainty.
The expectation of a stochastic process is a measure of the average or central tendency of the process. It represents the long-term average of the outcomes of the process and is calculated as the sum of the possible outcomes weighted by their respective probabilities.
Rewriting expectation is important in stochastic processes because it allows us to simplify complex expressions and make calculations more manageable. It also helps us to better understand the behavior of the process and make predictions about its future behavior.
Some common techniques for rewriting expectation in stochastic processes include linearity, independence, and conditional expectation. Linearity allows us to break down complex expressions into simpler ones by taking the expectation of each term separately. Independence allows us to treat random variables as if they are uncorrelated, making calculations easier. Conditional expectation is useful for finding the expectation of a random variable given information about another random variable.
Rewriting expectation in stochastic processes can be applied in various real-world situations, such as predicting stock prices, analyzing financial risks, and modeling the spread of diseases. It can also be used in engineering and physics to understand the behavior of systems that involve randomness, such as weather patterns or traffic flow.