Intereting Posts

Why is axiom of dependent choice necesary here? (noetherian space implies quasicompact)
Prove that for any piecewise smooth curve it is possible to find the parametrisation
Is there an easy way to show which spheres can be Lie groups?
Does the series $\sum \sin^{(n)}(1)$ converge, where $\sin^{(n)}$ denotes the $n$-fold composition of $\sin$?
Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs
Typo: in the definition of inverse image $E$ should be replaced by $H$
When is $X^n-a$ is irreducible over F?
Endpoint of a line knowing slope, start and distance
Clarifying a comment of Serre
Software for drawing geometry diagrams
exponentiation and modular arithmetic
Alternative definitions of stochastic processes?
Fourth Order Nonlinear ODE
Evaluating $\int_0^{\pi/2}({x \over \sin x})^2dx$ using value of a given integral
What about the continuity of these functions in the uniform topology?

Hello I’m having difficulties with the following problem:

Let X be a Poisson random variable with parameter $ \lambda $. Find the conditional mean of X given X is odd.

What I tried this:

A = [X is odd.]

- Everything in the Power Set is measurable?
- Borel $\sigma$-Algebra definition.
- Is this urn puzzle solvable?
- Show $\lim_{n \to \infty} \sum_{i=1}^n Y_i/\sum_{i=1}^n Y_i^2 = 1$ for Bernoulli distributed random variables $Y_i$
- In the card game Set, what's the probability of a Set existing in n cards?
- The Tuesday Birthday Problem - why does the probability change when the father specifies the birthday of a son?

Back to the basic idea

E[X|A] = $\displaystyle\sum\limits_{x=0}^\infty xP_{X|A}(X=x|A=a) = \displaystyle\sum\limits_{x=0}^\infty x \frac{P_{X,A}(X=x,A=a)}{P_{A}(A=a)} $

I actually don’t understand how the probability of X and X is odd is different from the probability of X is odd. I suppose the random variables X and A = [X is odd] are different, but if they are intersecting events how are they different in the end? My main confusion is that I don’t know how to differentiate between the joint probability and the probability of X being odd.

Thank you.

- Boy Born on a Tuesday - is it just a language trick?
- Probability that the student can solve 5 of 7 problems on the exam
- Magic 8 Ball Problem
- A subset $P$(may be void also) is selected at random from set $A$ and the set $A$ is then reconstructed by replacing the elements of $P$.
- Probability and uniform distribution
- What is the probability for a wood stick of real number length breaking in three piece that can forming precisely a triangle?
- The “Bold” strategy of a single large bet is not the best Roulette strategy to double your money
- Expectation (and concentration?) for $\min(X, n-X)$ when $X$ is a Binomial
- Does 0% chance mean impossible?
- Probability that a random binary matrix is invertible?

The idea is good. The conditional probabilities are a little bit off: recall that $\Pr(A|B)=\frac{\Pr(A\cap B}{\Pr(B)}$.

So we need to divide each of your terms, and hence your expression, by the probability that $X$ is odd. It is clear that you know how to compute that.

**Added:** The probability that $X$ is odd is

$$\sum_{i=0}^\infty e^{-\lambda} \frac{\lambda^{2i+1}}{(2i+1)!}.$$

This can be simplified a lot. Write down the power series for $e^\lambda$, also for $e^{-\lambda}$ Subtract. We get twice our sum. So the probability that $X$ is odd is $e^{-\lambda}\left(\frac{e^\lambda-e^{-\lambda}}{2} \right)$. Note this is $e^{-\lambda}\sinh \lambda$.

Now the probability that $X=2k+1$ **given** that $X$ is odd is, by the usual conditiona probability formula, equal to

$$\frac{e^{-\lambda}\frac{\lambda^{2k+1}}{(2k+1)!}}{e^{-\lambda}\sinh \lambda}.$$

For the conditional expectation, multiply the above expression by $2k+1$, and add up from $k=0$ to $\infty$. Since $\frac{2k+1}{(2k+1)!}=\frac{1}{(2k)!}$, we get, after a little manipulation, that the conditional expectation is

$$\frac{\lambda}{\sinh \lambda}\sum_{k=0}^\infty \frac{\lambda^{2k}}{(2k)!}.$$

By a calculation with the expansions of $e^\lambda$ and $e^{-\lambda}$ of the type we did before, or in another way, we can show that the inner sum is $\cosh \lambda$.

- Existence Proofs
- How do we know Taylor's Series works with complex numbers?
- Have there been (successful) attempts to use something other than spheres for homotopy groups?
- a Circle perimeter as expression of $\pi$ Conflict?
- Rational Numbers – LCM and HCF
- Closed form for $\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1)$
- Show that the set $M$ is not an Embedded submanifold
- Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula
- Prove this : $\left(a\cos\alpha\right)^n + \left(b\sin\alpha\right)^n = p^n$
- The group of roots of unity in the cyclotomic number field of an odd prime order
- How to prove $\lvert \lVert x \rVert – \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
- How do I solve this System of Equations?
- Using Axiom of Choice To Find Decreasing Sequences in Linear Orders
- Uniform convergence of derivatives, Tao 14.2.7.
- Are there more rational numbers than integers?