Intereting Posts

How are long proofs “planned”?
Tensor product of monoids and arbitrary algebraic structures
Showing the Lie Algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are not isomorphic.
Let $R$ be a commutative ring and let $I$ and $J$ be ideals of $R$. Show $IJ$ is an ideal of $R$.
Compact operator with closed range has finite dimensional range
Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has one root between $-1$ and $2$
Sufficient condition to inscribe a polygon inside another one
Conditional probability
Homology of some quotient of $S^2$
I need help finding a rigorous precalculus textbook
Is there a base in which $1 + 2 + 3 + 4 + \dots = – \frac{1}{12}$ makes sense?
Converting an ODE in polar form
Consistency strength of weakly inaccessibles without $\mathsf{GCH}$
Difficulties in a proof by mathematical induction (involves evaluating $\sum r3^r$).
IMO 1988 question No. 6 Possible values of $a$ and $b$, $\displaystyle\frac{a^2+b^2}{ab+1}$

Let T be a non-negative integer-valued random variable with $\mathbb{E}(T) < \infty $. Prove that

$\mathbb{E}(T) = \sum^\infty_{k=1}\mathbb{P}(T \geq k)$.

Had a few attempts, haven’t really got anywhere. I’m wondering as I’m typing this if proof by induction is a good way to go.

Edit: One major thing I forgot to add, am I correct in thinking that also, $\mathbb{E}(T) = \sum^\infty_{k=1}k\mathbb{P}(T = k)$?

- Should you ever stop rolling, THE SEQUEL
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- Prove that if expectations agree on a Pi-System, then they agree on the Sigma-Algebra generated by the Pi-System.
- Analysis of “Tiny Dice Dungeon” (I)
- Free throw interview question
- expected number of cards drawn exactly once (with replacement)

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- Generalization of variance to random vectors
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- Why are continuous functions not dense in $L^\infty$?
- Intuitive explanation of variance and moment in Probability
- Finite State Markov Chain Stationary Distribution

$$T=\sum_{k=1}^\infty\mathbf 1_{T\geqslant k}=\sum_{k=1}^\infty k\cdot \mathbf 1_{T=k}$$

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- Improper Integral $\int\limits_0^1\frac{\ln(x)}{x^2-1}\,dx$
- Identity and possible generalization of the reflective periodic continued fractions
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- Functions continuous in each variable
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- Exact same solutions implies same row-reduced echelon form?
- Why is the area under a curve the integral?
- Indicator function for a vertex-induced random subgraph of $G$?
- Why the $O(t^2)$ part in $L(t) = L + t(\csc \alpha_i – \cot \alpha_i +\csc \alpha_{i+1} – \cot \alpha_{i+1}) + O(t^2)$?
- Expected Value Of Dice Rolling Game
- Is there a continuous $f(x,y)$ which is not of the form $f(x,y) = g_1(x) h_1(y) + \dots + g_n(x) h_n(y)$
- The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
- Does $\cdot =$? Here $$ is the GIF function.