Intereting Posts

Is every countable dense subset of $\mathbb R$ ambiently homeomorphic to $\mathbb Q$
Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares
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$\sigma$ – compact and locally compact metric space
A convex optimization problem over two vectors
Open subschemes of affine schemes are affine?
$\tau$ and grouping of prime numbers
I want to study mathematics ahead of high school, but I found that I'm rusty on the elementary stuff
Lambert function approximation $W_0$ branch
Sum of these quotient can not be integer
An unexpected application of non-trivial combinatorics
Why do certain notations differ around the world?
Verify Gauss’s Divergence Theorem
Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$?
Formula for the sequence repeating twice each power of $2$

I am currently tasked with proving an alternative definition of the expected value function.

Considering X to be a random variable that takes all positive integers, I have to prove that

$E[X]=\sum\limits_{i=1}^{\infty} P[X\ge i]$.

So far I’ve gotten to the fact that $P[X\ge i] = \sum\limits_{j=i}^{\infty} P[X=j]$ however I just have no idea where to go from here. I have no idea how that relates to expected value in any way.

- To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
- Gamma Distribution out of sum of exponential random variables
- Expectation of $QQ^T$ where $Q^TQ=I$
- How do we identify a probability problem as a conditional probability problem?
- Probability of $(a+b\omega+c\omega^{2})(a+b\omega^{2}+c\omega)=1$
- What's the probability of a an outcome after N trials, if you stop trying once you're “successful”?

I’m most certainly not looking for the answer, but a friendly nudge in the right direction would be extremely helpful.

- What is a good strategy for this dice game?
- Correlation between three variables question
- Expected area of triangle formed by three random points inside unit circle
- What does multiplication mean in probability theory?
- Probability that n points on a circle are in one semicircle
- Limiting distribution.
- Help: rules of a game whose details I don't remember!
- Strong law of large numbers for uncorrelated $L^2$ random variables
- Variance of a stochastic process with Gaussian correlation function
- probability question (“birthday paradox”)

Let $p_i = P[X=i]$. You know that $$E[X]=\sum_{i=1}^\infty ip_i$$ and that $$P[X\ge i] = \sum_{j=i}^\infty p_j,$$ so you want to show that $$\sum_{i=1}^\infty ip_i = \sum_{i=1}^\infty \sum_{j=i}^\infty p_j\;.$$ Try reversing the order of summation in the double summation.

- RSA when N=pq and p = q
- “Visualizing” Mathematical Objects – Tips & Tricks
- Why this two spaces do not homeomorphic?
- Haar measure on SO(n)
- Extending an automorphism to the integral closure
- Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.
- Product of manifolds & orientability
- $X$ – regular, $A \subset X$ – closed $\Rightarrow \ \ X/A$ – quotient space is Hausdorff
- When is a Lipschitz homeomorphism of metric spaces bi-Lipschitz?
- Proving an operation of interior is a set of open sets.
- Markov property question
- Finitely generated ideal in Boolean ring; how do we motivate the generator?
- A local-global problem concerning roots of polynomials
- Does absolute convergence of a sum imply uniform convergence?
- Is there a prime number between every prime and its square?