Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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1answer
347 views

Maximum likelihood and fisher information of uniform and binomial

The MLE for a uniform distribution is at the corner and not where the FOC equal 0 (i.e. not where the derivative of the log likelihood equals 0) because the function is strictly monotonic. Hence, the ...
0
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1answer
41 views

Distribution of the minimum of a random sample

Suppose $x_i$ is a random variable with CDF $F(.)$ with a bounded support. I get a random sample $S_n=\{x_1,x_2,\dots,x_n\}$. Define $x_{\text{min}}=\min(S_n)$. How can I find the pdf of this random ...
4
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0answers
126 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
1
vote
1answer
74 views

Binomial thinning of geometric distribution

I need to show that a random variable follows a specific law, how could I do that? Let $\{A_{i,j}\}$ be an infinite iid array of Bernouilli($\psi$) variables and $\eta_1,\ldots,\eta_n$ be iid ...
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2answers
542 views

compound of gamma and exponential distribution

What is the distribution of a exponential distribution, whose parameter is drawn form the gamma distribution $ X \sim Gamma(\alpha,\beta)$ $ Y \sim Exp(X)$ how is $Y$ distributed? EDIT Since there ...
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1answer
95 views

Proving the equivalence of a pdf with a log-normal distribution

I have the following function normalised to 1 on $(0, \infty)$: $$ g(x) = \frac{e^{-\left(\mu + \frac{\sigma^2}{2}\right)} e^{-\frac{(\mu -\text{Log}[x])^2}{2 \sigma ^2}}}{\sigma \sqrt{2 \pi }} $$ ...
2
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1answer
792 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 $Y = \dfrac{1}{X}$
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1answer
61 views

Find the distribution of $Z=\min \{n: U_n \leq h(n) \}$

Let $X$ be a non-negative integer-valued random variable with probability mass function $f(k)=P(X=k)$ for $k=0,1,2,\ldots$ Define a function $h(r) = P(X=r | X \geq r)$. Let $U_i$ for $i=0,1,2,\ldots$ ...
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1answer
44 views

Proof of Convergence in Distribution and Limsup

I'm currently using 'Adventure in Stochastic Processes' for self-study. Here's the link. This is the part I don't understand: Letting $\left[n\rightarrow \infty\right]$, we get $\limsup_{n\rightarrow ...
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2answers
371 views

When does equality in Markov's inequality occur?

Markov's inequality states that given any nonnegative random variable and $a>0$ then we have: $$P(X \geq a) \leq \frac{E(X)}{a}$$ At which $a$ is equality supposed to hold?
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3answers
287 views

In how many ways can I distribute 6 identical cookies and 6 identical candies to 4 children, if each child must receive exactly 3 items?

I tried to solve this by making a chain of letters, with 'O' representing cookies and 'A' representing candies, as shown below. o o o o o o a a a a a a 1 1 1 2 2 2 3 3 3 4 4 4 This would mean that ...
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2answers
20 views

Finding Y's marginal distribution where joint distribution of $f_{X,Y}(x,y) = 1/2$ in $0 < x < 1$ and $0 < y < 4x$

I am given a two-dimensional vector (X,Y) whose joint density function is as follows: $f_{X,Y}(x,y)=1/2$ if $0<x<$ 1 and $0<y<4x$. I am now to find the marginal densities of X and Y. I ...
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2answers
309 views

pdf for non-central gamma distribution

I have a given gamma distribution as: $f(x;k,\theta) = \frac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{\frac{-x}{\theta}}$ and a non-centrality parameter $\delta$. Now, I need to find the pdf of this ...
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2answers
66 views

MGF/ expectation Gaussian Random Variables

I am stuck with something that seems easy but i cannot recall how to figure it out? Let $G_1$ and $G_2$ be two standard gaussian random variables with mean $0$ and variance $1$. Then how to calculate ...
3
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2answers
224 views

Characteristic Function and Random Variable Transformation

Let $X$ be a random variable, and let $\phi_X(t)$ be its characteristic function. Let $Y = f(X)$ be a transformation of the random variable $X$ where $f$ is increasing and one-to-one. Is there a ...
4
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0answers
88 views

Convergence in distribution of bernoulli rv over square root of uniform rv

This is a question from an old comprehensive exam: Let $U$ be a $\operatorname{Uniform}[0,1]$ random variable and let $X$ be a $\operatorname{Bernoulli}(1/2)$ random variable independent of $U$. ...
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1answer
72 views

How to show that two random proceses have the same family of finite dimensional distributions?

I got two random processes: $$y_t=e_t-\frac{1}{3}e_{t-1},\ e_t\sim\mathcal{N}(0,9)\ \text{i.i.d.}$$ $$y_t=e_t-3e_{t-1},\ e_t\sim\mathcal{N}(0,1)\ \text{i.i.d.}$$ I want to show that both have the ...
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1answer
61 views

Cumulative Distribution Function Proof

So I have this proof formula for the Cumulative Distribution Function and I understand it up till the point where we see square brackets, how did those end up there and how did division ended up ...
0
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1answer
26 views

Sampling distribution of an estimator

All random variables here are iid. We have that $$f(x;\theta) = \alpha \mbox{ exp}\left[ -\alpha (x - \beta) \right] \times \mathbb{I} \left\{x \geq \beta \right\} $$ for $\alpha > 0$ and $\beta ...
2
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1answer
69 views

How to prove that if $\int_t^\infty(s-t-\frac{1}{\lambda})\,f(s)\ ds =0$ for all $t\ge 0$ then $f(s)=\lambda\, \mathrm{e}^{-\lambda s}$

The problem is motivated by my probability text which states that if the expectation of time to wait conditioned on time already spent waiting $(t)$ is constant (equals $\frac{1}{\lambda}$) then the ...
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1answer
111 views

Lebesgue integral of a bounded random variable

Given a random variable $X$, if we take a measurable and bounded function $f(X)$ then can we say that $f$ is Lebesgue integrable wrt a probability measure on $\mathbb R$? In Real Analyses book by ...
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2answers
97 views

how to solve this integral in survival analysis

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
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2answers
213 views

Distribution of angle of two dimensional normal vector

The original subject is: Suppose random variables $X$ and $Y$ are independent and both follow the Normal distribution $N(0,\sigma ^2)$. 1) Prove $U=X^2+Y^2$ and $V = \frac{X}{\sqrt{X^2+Y^2}}$ ...
0
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1answer
80 views

How do I fit a gamma distribution to a pdf measured by maximum likelihood.

I am terrible at statistics, nonetheless I was given the task of trying to figure out how to fit our company's data to a distribution using Maximum Likelihood Estimation. After a very long time ...
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2answers
27 views

pdf vs cdf -random variables-

The following problem is what I am working on. $F(x)=\frac{1}{1+e^{-x}}$ is the cumulative density function defined for all real numbers. Find the probability density function. My understanding ...
2
votes
2answers
147 views

Standard Deviation greater than mean implies no normal distribution?

I understand that the mean $\mu \pm \sigma$ gives a better approximation of the measurements. But how is it related to the normal distribution? Is it because since $\sigma > \mu$ so the normal ...
0
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1answer
85 views

Suppose that $X_1, X_2, …, X_n$ are i.i.d. random variables such that $X_1\sim N(\mu, 0.5)$ and $n = 100$.

Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d. random variables such that $X_1\sim N(\mu, 0.5)$ and $n = 100$. a) Find a maximum number $c$ such that $$P(X_1\le c+\mu, X_2\le c+\mu, \ldots,X_n\le ...
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0answers
70 views

A naive question about “random” probability distributions

So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable $X$ taking values in the interval $[-1,1]$. Then, say, ...
1
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1answer
119 views

Exponential family of distributions?

Consider the parametric class formed by the density functions defined as follows: $$ f(y,\theta) = \frac {2} {\Gamma (1/4)} e^{-(y-\theta)^4},\quad y\in\mathbb R,\quad\theta\in\mathbb R. $$ Does ...
0
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3answers
481 views

exponential distribution expected value of all values greater than some value

Mean of exponential distribution is $$\frac{1}{\lambda}$$ What is mean of all samples greater than some value $S$? Some context: A DC power supply (as used in a telecommunications installation) has ...
3
votes
1answer
124 views

Is it possible to obtain the Uniform distribution as the difference of two independent random variables?

Is it possible to have two independent random variables X,Y with identical distribution, such that $X-Y \sim \text{Uniform}[a,b]$? I am almost certain that is not, but maybe I am overlooking ...
2
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2answers
329 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
1
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1answer
79 views

A question about stochastic ordering and convolution

Two probability density functions $f$ and $g$ are known to have distribution functions $F$ and $G$ respectively with $F(y)>G(y)$ for all $y$, say on $\mathbb{R}$. It is known that if we convolve ...
5
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1answer
123 views

Log-likelihood function

I'm not sure if this could be asked here, or in math overflow... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
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2answers
120 views

Finding expected value of conditional probability distribution

Suppose we throw a fair dice, and let $X$ be the random variable representing each face (each result: $1,2,3,4,5,6$). Suppose that we drew $k$ on the dice; we then flip a fair coin $k$ times. let $Y$ ...
2
votes
1answer
34 views

probability of arbitrary distribution that value is $> \mu+/-3\sigma$

A normal distribution has the property: $P(X>\mu+/-3\sigma)=1-99.71$ What is the probability $P(X>\mu+/-3\sigma)$ for an arbitrary distribution? (Is it actually possible to have some general ...
0
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1answer
35 views

A very basic question about the Boltzmann distribution

I understand the formula for the Boltzmann distribution to be $P(E_i) = e^{-E_i/(kT)}/Z$ When the energy levels vary continuously illustrations of pdf for either the energy or the velocity at a ...
4
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2answers
91 views

Intuitive way to arrive at the maximizing argument for the binomial probability

The binomial probability term $q^{n}(1-q)^{N-n}$ is maximized when $q=n/N$. This can be easily arrived at by differentiating the given probability term with respect to q. Is there a more intuitive way ...
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2answers
586 views

Does the weibull distribution has a sufficient statistic?

When using the following definition of weibull: $f(y) = \beta \alpha y^{\alpha - 1}e^{-\beta y ^ {\alpha}} $ , When $\beta>0 \alpha >0$. I could only find (using the factorization theorem) ...
0
votes
1answer
18 views

Probability and Statistics Expected Weekly Loss

So I have this exercise (picture below) and I have the mean calculated and the variance of the random variable X, so I'm left with the formula for Loss, should I simplu substitute the X in the E(10X ...
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1answer
24 views

The time to hatch between $k_1$ and $k_2$ eggs (if hatching times are independent and exponentially distributed)

I have $N$ eggs that each hatch after a time given by independent exponentially distributed random variables with identical rate parameters $\lambda \space seconds^{-1}$. At our initial time $t = 0$, ...
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2answers
156 views

Given four independent random variables $X_1,\cdots,X_4$ find $Y=\min\{X_1,X_2,X_3,X_4\}$

Suppose that we're given four independent random variables $X_1,X_2,X_3$ and $X_4$ and their probability density function is given by: $f(x)= 3(1-x)^2 $ for $0<x<1$ and otherwise $f(x)=0$. If ...
0
votes
1answer
26 views

name of plot for discrete joint probability (table) visualizing black squares of varying size per cell

I was wondering what is the name of a specific plot type used for discrete joint probabilities. This plot has a table structure and each cell is a black square of varying size, with size proportional ...
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2answers
65 views

Test for, and compare means of folded normal distribution

I have two datasets of absolute distances to a single point in a 2D space. I have reasons to expect that if I had the sign and magnitude of these distances, my datasets would be normally distributed ...
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1answer
100 views

Source for Probability of one point in Distribution is zero

I know from my old studies that the probability of one single point, for instance in a normal distribution and quasiprobability distribution, is zero. Where should you cite for this fact? I have ...
1
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1answer
62 views

PDF of summation of two random variables (different than uniform)

My question is different than the questions about uniform distribution. Question: X and Y have the following pdf: f(x,y) = (1/4)*x*y if 0 < x < 2 and 0 < y < 2 f(x,y) = 0 otherwise ...
2
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1answer
629 views

Central Limit theorem Application on Poisson Distribution

Suppose that $X_1,\ldots,X_n$ is an iid sample from the Poisson distribution with mean $\lambda$. Use the Central Limit theorem to find $P(|\bar X - \lambda| < 0.1) $ as $n$ goes to infinity. My ...
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0answers
57 views

probability density of a real symmetric random matrix

I learn from Steve Lalley's lecture notes that the joint probability density of the entries of a real symmetric random matrix is a function only of the eigenvalues of the matrix. Then, let $RR^T$ be ...
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votes
1answer
43 views

How can I deal with it? I want to show $X$ and $Y$ are independent.

I simply cannot deal with it. how to start.... the problem follows : Let $X$ and $Z$ be independent with $X \sim N(0,1)$ and $P(Z=1)=P(Z=-1)=\frac{1}{2}$. Let $Y=X \times Z$. Prove: (a) $Y \sim ...
1
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2answers
3k views

Expected value of two dependent variables is still a product of expectations

For independent variables we have $E[XY]=E[X]E[Y]$. Now, since I could not find a statement that the converse is also true, I suspect that there are examples of dependent variables where this relation ...