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Results tagged with cumulative-distribution-function
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user 54458
Cumulative distribution function. While the PDF gives the probability density of each value of a random variable, the CDF (often denoted $F(x)$) gives the probability that the random variable will be less than or equal to a specified value.
1
vote
If $X=\sin\Theta$ and $Y=\cos\Theta$ with $\Theta$ uniformly distributed, how can I compute ...
$\Theta \sim U(-\pi, \pi)$ so the density of $\Theta$ is given by $\frac{1}{2 \pi}$ in $-\pi, \pi$.
$F(x, y) = P(X \le x, Y \le y) = P(\Theta \le \arcsin(x) \wedge \arccos(y))$.
$
F(x, y) = \frac{1 …
- 2,238
46
votes
Accepted
Why is the CDF of a sample uniformly distributed
Assume $F_X$ is continuous and increasing. Define $Z = F_X(X)$ and note that $Z$ takes values in $[0, 1]$. Then
$$F_Z(x) = P(F_X(X) \leq x) = P(X \leq F_X^{-1}(x)) = F_X(F_X^{-1}(x)) = x.$$
The deriva …
- 2,238