# Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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### Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
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### Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X

Let $X$ be a random variable with a continuous and strictly increasing c.d.f. function $F$ (so that the quantile function $F^{−1}$ is well-deﬁned). Deﬁne a new random variable $Y$ by $Y = F(X)$. Show ...
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### Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $(\Omega, \mathscr{A}, \mu )$ ...
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### Intuition behind Variance forumla [duplicate]

Variance is given as: $\operatorname{Var}(X) = \mathbb{E}[(X-\mathbb{E}(X))^2]$. Is there an intuition behind this and can you find this formula starting from the second generating moment ?