Expected values in Probability space

In summary, the conversation discusses finding an example where I_{p}(lim inf_{n -> ∞}X_{n}) < lim inf_{n -> ∞}I_{p}(X_{n}). The attempt at a solution suggests creating a non-convergent sequence Xn with constant expected values, and evaluating the integrals to find lim inf Xn and E[Xn]. However, there is uncertainty in evaluating the integrals and further guidance is needed.
  • #1
Lily@pie
109
0

Homework Statement



Let a probability space be [itex](Ω, \epsilon, P)[/itex]. A set of random variables X1,...,Xn

Give an example where [itex]I_{p}(lim inf_{n -> ∞}X_{n}[/itex]) < [itex]lim inf_{n -> ∞}I_{p}(X_{n})[/itex]


The attempt at a solution

I know that [itex]I_{p}(lim inf_{n -> ∞}X_{n}[/itex])=[itex]E[lim inf_{n -> ∞}X_{n}[/itex]]
and [itex]lim inf_{n -> ∞}I_{p}(X_{n})[/itex] = [itex]lim inf_{n -> ∞}E[X_{n}][/itex]

I think I need to find a sequence of Xn such that lim inf Xn will have a smaller value than all the individual expected value, E[Xn].

Am I on the correct path? I'm kind of stuck here and not sure how to proceed.

Would be really really thankful for the help.
 
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  • #2
How is ##\displaystyle \liminf_{n \to \infty} X_n## defined? The limit of the "smallest value" of those random variables?
In that case, I don't see how both can be equal apart from trivial Xi.
 
  • #3
Equality actually holds for surprisingly many cases. See for example, the dominated convergence theorem. So you'll need to look at sequences of random variables which fail that theorem.
Without giving away too much, try to make a nonconvergent sequence ##X_n## such that ##E[X_n]## are all constant.
 
  • #4
Okay, then I don't understand how to evaluate ##\displaystyle \liminf_{n \to \infty} X_n##.
 
  • #5
If ##X = \liminf_n X_n##, and ##\omega## is an outcome, then it is defined by
[tex]X(\omega) = \liminf_n X_n(\omega)[/tex]

So it's just the pointswise inferior limit.
 
  • #6
micromass said:
try to make a nonconvergent sequence ##X_n## such that ##E[X_n]## are all constant.

I saw this example...

Ω=N
P(ω)=2, ω in Ω
Xn(ω)=2nδω,n

I can see that Xn is not convergent.

But I'm not quite sure in computing the integrals... I'm very bad with the lim inf concept >_< So, I have written my attempt on evaluating the integrals and reasoning that I've used.

My attempt,
lim inf Xn
=lim inf [2nδω,n]
Since this takes in values 0 or 2n
=0

E[Xn]
=IP[2nδω,n]
= ∫2nδω,n2
When ω=n, this reduces to ∫δω,n
ω,n

I'm not very sure about the way I evaluate the integral. Would be very helpful for some guidance.
 

What is an expected value?

An expected value is a predicted outcome of a random variable in a probability space. It represents the average result that can be expected if the experiment or process is repeated multiple times.

How is the expected value calculated?

The expected value is calculated by multiplying each possible outcome of a random variable by its probability, and then summing all of these products together. This calculation takes into account the likelihood of each outcome occurring.

What is the significance of the expected value?

The expected value is significant because it allows us to make predictions and decisions based on probabilities. It provides a measure of central tendency that can be used to compare different outcomes and assess the potential risks and rewards of a given situation.

Can the expected value be negative?

Yes, the expected value can be negative if the potential outcomes of a random variable include negative values. This can happen when there is a higher probability of a negative outcome compared to a positive one.

Can the expected value be greater than the maximum possible outcome?

Yes, the expected value can be greater than the maximum possible outcome if the potential outcomes of a random variable include very large values with a low probability of occurring. In these cases, the expected value is a hypothetical average that may not be the most likely outcome.

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