query: Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
positive_passages.docid: 16673
positive_passages.text: If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.
negative_passages.docid: 48270
negative_passages.text: How to prove that the random variables $X$ and $X^2$ are not independent?
negative_passages.docid: 52095
negative_passages.text: Prove either $G=ST$ or |$G|\geq|S|+|T|$
negative_passages.docid: 56413
negative_passages.text: $f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
negative_passages.docid: 110313
negative_passages.text: Prove that $\gcd(f(x),g(x)) = 1$.
negative_passages.docid: 70912
negative_passages.text: If $g\circ f$ is injective and $f$ is surjective then $g$ is injective
negative_passages.docid: 70128
negative_passages.text: Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
negative_passages.docid: 85612
negative_passages.text: Let functions $ f: A \to B$ and $g: B \to A$, suppose that $ g \circ f = i_A$. Prove that if $f$ is onto, then $g$ is one-to-one
negative_passages.docid: 138343
negative_passages.text: How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent?
negative_passages.docid: 152495
negative_passages.text: Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent
negative_passages.docid: 131357
negative_passages.text: Prove that $f(x,y)=xy$ is differentiable using the definition
negative_passages.docid: 54461
negative_passages.text: Showing independence of random variables
negative_passages.docid: 96354
negative_passages.text: Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)
negative_passages.docid: 18967
negative_passages.text: Linear independence of $\sin(x)$ and $\cos(x)$
negative_passages.docid: 57023
negative_passages.text: If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
negative_passages.docid: 144589
negative_passages.text: Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
negative_passages.docid: 82935
negative_passages.text: Prove that f∈Θ(g)if and only ifO(f) =O(g).