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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.

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21
votes
3answers
8k 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
20k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
27
votes
3answers
14k 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\...
28
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 ...
34
votes
4answers
19k 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 \...
45
votes
7answers
117k 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{...
25
votes
5answers
41k 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 ...
37
votes
1answer
8k 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 ) $ ...
35
votes
5answers
50k 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}...
2
votes
2answers
4k 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 ...
14
votes
2answers
29k 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 $0$ ...
13
votes
3answers
30k 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 ...
10
votes
3answers
3k 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 ...
9
votes
1answer
8k 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.
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 ...
6
votes
3answers
479 views

Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$

It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$ Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$ I was thinking of ...
25
votes
1answer
12k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent Gaussian random variables. The trouble is, my Gaussian random variables are not independent. ...
18
votes
3answers
35k 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 ...
16
votes
2answers
34k 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
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 ...
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{\...
56
votes
5answers
71k 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 ...
19
votes
2answers
8k 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 ...
17
votes
2answers
31k 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)
12
votes
3answers
30k 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 ...
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
734 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 $$\mathbb{E}[\phi(X)] = \int_{-\infty}^\infty \phi(x) f(x)dx$$ ...
12
votes
1answer
6k views

Distribution of the digits of Pi

Can anything be stated about the distribution of the digits of Pi, i.e., if I were to sample n digits of Pi, can anything be said about the probability to observe certain digits, or is there any ...
6
votes
3answers
1k 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 ...
21
votes
2answers
20k 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 ...
13
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{...
10
votes
2answers
17k 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} \>, \...
10
votes
2answers
1k 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 ...
6
votes
1answer
17k views

Tossing a fair coin until two consecutive tosses are the same

A fair coin is tossed repeatedly and independently until two consecutive heads or two consecutive tails appear. What is the PMF of the number of tosses?
5
votes
1answer
15k 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 $Z=X-Y$?
2
votes
1answer
390 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 ...
5
votes
3answers
20k 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, ...
24
votes
3answers
70k 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 ...
9
votes
3answers
30k views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
12
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?
14
votes
1answer
11k 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 ...
7
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 ...
4
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 ...
8
votes
3answers
2k views

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 ?
7
votes
2answers
17k 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
850 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)...
1
vote
4answers
15k views

Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial

The problem: X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$ Expected value is $\sum{x_ip(x_i)}$ But ...
9
votes
2answers
18k 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+...
2
votes
2answers
5k 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=...