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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.

21
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3answers
7k views

How to deduce the CDF of $W=I^2R$ from the PDFs of $I$ and $R$ independent

Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$? Where, $$ \begin{align} f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\ f_R(r)&=2r, &0 \leq r\...
23
votes
2answers
19k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
25
votes
3answers
13k views

Expectation of the maximum of i.i.d. geometric random variables

Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is $$E_n = E\left(\max_{i \in 1 .. n}X_n\...
26
votes
3answers
19k views

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 ...
32
votes
4answers
17k views

Why does the median minimize $E(|X-c|)$?

Suppose $X$ is a real-valued random variable and let $P_X$ denote the distribution of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP_X(x). $$ The medians of $X$ are defined as any number $m \in \...
21
votes
5answers
36k views

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 ...
35
votes
1answer
7k views

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 ) $ ...
44
votes
7answers
106k views

Poisson Distribution of sum of two random independent variables $X$, $Y$

$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{...
2
votes
2answers
3k views

$P(X>0,Y>0)$ for a bivariate normal distribution with correlation $\rho$

$X$ and $Y$ have a bivariate normal distribution with $\rho$ as covariance. $X$ and $Y$ are standard normal variables. I showed that $X$ and $Z= \dfrac{Y-\rho X}{\sqrt{1-\rho^2}}$ are independent ...
8
votes
1answer
7k views

If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.

Knowing that if you have two independent $X$ and $Y$, and $ f $ and $ g $ measurable functions, how to show that then $ U = f (X) $ and $ V = g (Y) $ are still independent.
9
votes
3answers
2k views

probability distribution of coverage of a set after $X$ independently, randomly selected members of the set

I have a set of numbers where I am randomly and independently selecting elements within a set . After a number of these random element selections I want to know the coverage of the elements in the ...
16
votes
3answers
32k views

Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
13
votes
3answers
26k views

Proof that the sum of two Gaussian variables is another Gaussian

The sum of two Gaussian variables is another Gaussian. It seems natural, but I could not find a proof using Google. What's a short way to prove this? Thanks! Edit: Provided the two variables are ...
13
votes
2answers
31k views

How to compute the sum of random variables of geometric distribution

Let $X_{i}$, $i=1,2,\dots, n$, be independent random variables of geometric distribution, that is, $P(X_{i}=m)=p(1-p)^{m-1}$. How to compute the PDF of their sum $\sum_{i=1}^{n}X_{i}$? I know ...
5
votes
3answers
7k views

X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)

Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together. The exercise goes as ...
8
votes
3answers
8k views

CDF of a ratio of exponential variables

Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$. I tried out the problem, and wanted to check to see if my answer of: $\frac{\...
5
votes
1answer
6k views

Characteristic function of product of normal random variables

I would like to find the characteristic function of the product of two independent brownian motions. This boils down to the characteristic function of the product of two normal random variables. This ...
54
votes
5answers
66k views

How can a probability density be greater than one and integrate to one

Wikipedia says: The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one. and it also says. Unlike a probability, a probability density ...
31
votes
4answers
47k views

Expected Value of a Binomial distribution?

If $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom nkp^k(1-p)^{n-k}...
13
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2answers
28k views

Sum of two uniform random variables

I am calculating the sum of two uniform random variables $X$ and $Y$, so that the sum is $X+Y = Z$. Since the two are independent, their densities are $f_X(x)=f_Y(x)=1$ if $0\leq x\leq1$ and $0$ ...
9
votes
2answers
10k views

Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
5
votes
4answers
654 views

Conditional expectation of independent variables

Claim. Let $Z_1, Z_2$ be two independent and identically distributed random variables. Then we have: $$ \mathbb E[Z_1|Z_1+Z_2] =\frac{Z_1+Z_2}{2}. $$ Proof. To see this, I have proceeded as follows. ...
4
votes
1answer
2k views

What is the name of this theorem, and are there any caveats?

For random variable $X$ that follows some distribution, $f(x)$ is the probability density function of that distribution if and only if $$\mathbb{E}[\phi(X)] = \int_{-\infty}^\infty \phi(x) f(x)dx$$ ...
6
votes
3answers
962 views

Distribution of $(XY)^Z$ if $(X,Y,Z)$ is i.i.d. uniform on $[0,1]$

$X,Y$ and $Z$ are independent uniformly distributed on $[0,1]$ How is random variable $(XY)^Z$ distributed? I had an idea to logarithm this and use convolution integral for the sum, but I'm not sure ...
19
votes
2answers
7k views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
12
votes
1answer
4k views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ $$\text{...
12
votes
3answers
28k views

How exactly are the beta and gamma distributions related?

According to Wikipedia, the Beta distribution is related to the gamma distribution by the following relation: $$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$ Can you point me to a derivation of this ...
5
votes
1answer
13k views

Distribution of a difference of two Uniform random variables?

Let $X$ and $Y$ both be distributed between $[1,2]$, what is the distribution of $Z=X-Y$?
4
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3answers
18k views

The mode of the Poisson Distribution

Lately, I am doing an investigation on Stirling's formula and its applications. So I thought I could use it to prove that the mode of the Poisson model is approximately equal to the mean. Of course, ...
2
votes
1answer
223 views

Why Sampling without replacement gives better CI performance?

I was learning confidence intervals progressing slowly with few hiccups 1, 2, and wrapping up while found few more issues, one of which I have detailed here. Requesting your kind help. I created a ...
21
votes
2answers
19k views

Order statistics of i.i.d. exponentially distributed sample

I have been trying to find the general formula for the $k$th order statistics of $n$ i.i.d exponential distribution random variables with mean $1$. And how to calculate the expectation and the ...
21
votes
3answers
63k views

Sum of independent Gamma distributions is a Gamma distribution

If $X\sim \mathrm{Gamma}(a_1,b)$ and $Y \sim \mathrm{Gamma}(a_2,b)$, I need to prove $X+Y\sim(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and ...
11
votes
1answer
5k views

Sum of Independent Folded-Normal distributions

Let $X$ and $Y$ be independent, normally distributed random variables. How is $|X| + |Y|$ distributed? Is it known to be $|Z|$, where $Z$ is distributed normally?
10
votes
2answers
16k views

Derivation of chi-squared pdf with one degree of freedom from normal distribution pdf

How can we derive the chi-squared probability density function (pdf) using the pdf of normal distribution? I mean, I need to show that $$f(x)=\frac{1}{2^{r/2}\Gamma(r/2)}x^{r/2-1}e^{-x/2} \>, \...
12
votes
1answer
10k views

Distribution of Difference of Chi-squared Variables

I am trying to get the probability distribution function of $Z=X-Y$. Given that $f_X(x)$ and $f_Y(y)$ are known, and both variables are chi-square distributed, $X\in\mathbb{R}$, $X\ge 0$, and ...
15
votes
2answers
28k views

How to Prove that the minimum of two exponential random variables is another

How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y)
6
votes
2answers
5k views

Probability of $n$ successes in a row at the $k$-th Bernoulli trial… geometric?

If one has Bernoulli trials with success probability $p$, then it makes sense that the probability of the first success observed to be at trial number $k$ be given by $$(1-p)^{k-1} p.$$ But how ...
8
votes
2answers
16k views

Proof that Conditional of Poisson distribution is Binomial

The classic example... $X \sim Po\left (\lambda\right ), Y \sim Po\left (\mu\right)$, X and Y are independent. Show that the conditional distribution of X is binomially distributed. Or in other words,...
6
votes
0answers
814 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} f(x)p(x)...
8
votes
2answers
17k views

Distribution of the sum of squared independent normal random variables.

The sum of squares of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then $X_1^2+...
7
votes
1answer
6k views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
2
votes
2answers
4k views

The distribution of the minimum of two independent geometric random variables

Let $X$ and $Y$ be independent geometric random variables. What is the distribution of $Z=\min(X,Y)$? The probability mass functions are $\operatorname{Pr}(X=k)=(1-p)^{k-1}p$ and $\operatorname{Pr}(Y=...
0
votes
3answers
2k views

density of $X^2$ when $X$ has uniform $[-1, 2]$ distribution

Suppose $X$ has uniform $[-1,2]$ distribution. I am trying to find the density of $Z=X^2$. Here is what I have done thus far: Range($Z$)$=[0,4]$. I began computing the distribution of $Z$ for $z \in ...
1
vote
1answer
736 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle P(Z=...
23
votes
3answers
2k views

Existence of independent and identically distributed random variables.

I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know ...
9
votes
2answers
887 views

Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing

Let $X$ be a random variable and let $g,f$ be real-valued, nondecreasing, and bounded. Show that $\mathbb{E}[f(X)g(X)]\geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ Having a hard time seeing where to start ...
3
votes
2answers
2k views

Limit distribution of infinite sum of Bernoulli random variables

I know that the finite sum of Bernoulli i.i.d. random variables is a binomial distribution, but what is the distribution of $$\lim_{n \to \infty}\sum_{k=1}^{n} \frac{x_k}{2^k}$$ where $x_k$ is a ...
23
votes
3answers
2k views

Probability of picking an odd number from the set of naturals?

I know there's no uniform distribution for a countably infinite set, but I'm wondering if there's still a way to determine the probability of picking from a subset of a countably infinite set. For ...
7
votes
5answers
11k views

Negative binomial distribution - sum of two random variables

Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$ How do I go about proving this? I'm not sure where to begin, I'd be glad for ...
7
votes
2answers
6k views

Finding the moment generating function of the product of two standard normal distributions

The following question is on my homework assignment that I cannot figure out: Let U and V be independent random variables, each having a normal distribution with mean zero and variance one. Find ...