# Tagged Questions

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### $|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$

Let $a,b,c ∈ \Bbb R$ Show that $|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$ Not sure where to start
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### Prove that the following function of binary random variables is monotonic

Consider a binary random variable $y$ over the space $\mathcal{Y} = \{+1, -1\}$ such that $\Pr(y = 1) = q$. Consider also $r$ binary random variables $y^1, \ldots, y^1$ over the space $\mathcal{Y}$ ...
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### Sum of absolute values and the absolute value of the sum of these values?

I'm working on a proof and I need some help with this: I determined that for some situations ($x$ or $y$ are negative but not both): $|x| + |y| > x + y$ How can I conclude using that statement ...
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### Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
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### How does one DERIVE the formula for the maximum of two numbers

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
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### Prove $||a| - |b|| \leq |a - b|$ [duplicate]

I'm trying to prove that $||a| - |b|| \leq |a - b|$. So far, by using the triangle inequality, I've got: $$|a| = |\left(a - b\right) + b| \leq |a - b| + |b|$$ Subtracting $|b|$ from both sides yields, ...
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### Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
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### Adding equations in Triangle Inequality Proof

Inequality to prove: $|a+b|\leq |a| + |b|$ Proof: $-|a| \leq a \leq |a|$ $-|b| \leq b \leq |b|$ Add 1 and 2 together to get: $-(|a|+|b|)\leq a+b\leq|a|+|b|$ $|a+b|\leq|a|+|b|$ What is the ...
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### Proving square root of a square is the same as absolute value

Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't ...
I need help proving the last two cases for the following inequality: $\bigl|\lvert x\rvert-\lvert y\rvert\bigr| \le \lvert x-y\rvert$. Case 1: $x > 0$ and $y > 0$: the inequality simplifies ...
### Prove by contradiction or contrapositive? If $|x+y|<|x|+|y|$, then $x<0$ or $y<0$.
Prove: If $|x+y|<|x|+|y|$, then $x<0$ or $y<0$ This looks as though it's true from the start. Take $x=-4, y=4$. $|-4+4|<|-4|+|4|$ $0<8$ is true. The question is asking for a ...