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

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approximating a probability density

Let $f(x)$ be the probability density of a random variable $X$. Let the support of $f(x)$ be positive reals. If $f(x)$ is sufficiently smooth then one can approximate it with its Taylor series cut off ...
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171 views

Compare two estimators by using the their Expected value and variances

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
166 views

Joint and marginal distributions of independent uniformly distributed variables

Suppose that $X_1$ and $X_2$ are independently uniformly distributed on the interval (0,1). Find the joint and marginal distributions of $U=X_1X_2$ and $V=X_1/X_2$. I think that $f_U(u) = ...
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159 views

Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distrib… Help find finding $ \text{Var}\left[\hat{\theta}_{2}\right]$

Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Find the efficiency of $θ^1$ relative ...
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85 views

Finding density function of random variable, which is division of two other random variables.

I have following 2-dimensional random variable $(x,y)$: $$ f(x,y) = 1, \quad 0 \leq x \leq 1, \quad 0 < y \leq 1 $$ I have to find density function of random variable $Z = \frac{X}{Y}$. I am ...
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30 views

Sum of distribution

If $X_i \sim Bernoulli(\theta)$, then the $\sum_1^n X_i \sim Binomial(n, \theta)$ I don't know how it is derived, could anyone show or prove it to me. Besides, there are similar knowledges such as ...
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1answer
102 views

Finding the MLE for $θ$ given a probability density function $f(y |\theta )$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
30 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
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1answer
57 views

Show that if for $n \ge 1$, $(X_0, \ldots, X_n)$ has probability function $f_n$, then $X$ is a Markov Chain with transition matrix $\mathbb P$

Let $S$ a set of states and $\mathbb P=\{p_{i,j}\}_{i,j \in S}$ be a transition matrix. I've proved that $f_n(i_0,\ldots,i_n) := f_0(i_0) p_{i_0 i_1} \dots p_{i_{n-1} i_n}, \ \ (i_0, \ldots, i_n) ...
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$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find an estimator for $θ$ by the method of moments.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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2answers
374 views

Show that $ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$ is unbiased estimators of $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Show that $\hat{\theta}_2$ is ...
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1answer
35 views

Show that for $0<k<1$ $P(k < \frac{Y_{(n)}}{\theta} \le 1) = 1 - k^{cn}$.

The distribution function for a power family distribution is given by $$F(y)=\begin{cases} 0, & y<0\\ \left(\frac{y}{\theta}\right)^\alpha, &0\le y \le \theta \\ 1, ...
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2answers
43 views

Probability of 3 dices

Been looking through past exam papers and came across this question: Three fair dices are rolled. The probability that all three dices show 5 is 1/216. Is this true?
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82 views

Finding the C.D.F. from a P.D.F.

Suppose that X is a random variable that has a p.d.f. given by the formulas p(x)=a⋅(1−x3) for 0≤x≤1 and p(x)=0 for all other x, where a>0 is a constant. Find a formula for the c.d.f. FX(x) when 0≤x≤1. ...
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1answer
237 views

Probability Distribution of Runs in Coin Flips

If you flip a coin $n$ times, what is the probability distribution of the longest "run" (sequence of consecutive heads or tails) which will occur? Or if that's not possible, what is the average? I ...
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2answers
46 views

Deck of playing cards

Been going through an previous exam question and came across this: 5 cards are drawn from a deck of playing cards. What is the probability of drawing 3 aces? How do you calculate it using the C(n,r)? ...
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0answers
124 views

How's the damping factor in Google PageRank algorithm calculated

I'm doing some researches about Google's PageRank algorithm for my thesis, I've found that the damping factor x (for example), where x is in : P` = x.P + (1-x)Q where P is the original ...
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1answer
47 views

relation between multivariate probability generating function and univariate ones

Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I ...
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Distribution of the product of Gaussian column matrix and Signed Bernoulli matrix

This is not a homework question though it might be trivial for those who are well versed with multivariate distributions. I am trying to understand a paper that has the following product form. Let $A$ ...
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1answer
44 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
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1answer
72 views

multivariate probability generating function

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose ...
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1answer
26 views

How to find the distribution of X from the moments of X?

For example the moments of X is defined by E(x^n)=0.7, for $n\in[1,\infty]$. How to get the distribution of X?
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95 views

Derive minimum length confidence bounds for a F distribution variance …

Derive minimum length confidence bounds for a F distribution variance $\sigma^2$ and the ratio of two F distribution population variances $\frac{\sigma_1^2}{\sigma_2^2}$. What I got so far is $$ ...
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3answers
90 views

Given $X$ and $Y$ are independent N(0,1) random variables and $Z = \sqrt{X^2+Y^2}$ from the marginal pdf of $Z$

Let $X$ and $Y$ be independent $N(0; 1)$ random variables. Let $Z = \sqrt{X^2+Y^2}$. (a) Derive the marginal pdf of $Z$ and then using the marginal pdf to compute ${\rm E}[Z^2]$ (b) Can you propose ...
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1answer
105 views

LR Test for Exponential Family of Distributions

LR Test for Exponential Family of Distributions: The exponential family of distributions, both discrete and continuous, based on a parameter θ is defined by: f (x |theta) = ...
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1answer
20 views

Joint p.d.f and independence

I kinda remember there is a result like this from Probability theory, but I forgot how to prove it. Is there a formal name for it? Can someone kindly provide me with the proof or a link please The ...
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1answer
39 views

Finding the distribution function (homework help)

I do not quite understand what is classified as a distribution function and how to find a density function if one is a distribution function. For example how can I determine if the following is a ...
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39 views

Maximum likelihood to throw exactly two 6s

One throws a dice $n$ times. For which value of $n$ is maximum the probability to obtain exactly two 6s? I get $$n=11 \text{ or } n=12.$$ My solution: the probability to obtain exactly two 6s in ...
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voronoi graph generated by $k$-means clustering

assume I correctly find the optimal centroids $c_i$ in the kmeans clustering problem, which finds $k$ centroids that minimizes: $$ \min \sum_i \sum_{x_j\in C_i } \|x_j - \mu_i\|^2 $$where $\mu_i, \ ...
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1answer
48 views

Distribution of a uniform random variable with random endpoint

Let $Y \sim U[0,k]$, where $0 < k < \infty$ and $U$ is a continuous uniform distribution. Now let $X \sim U[0, Y]$. What is the distribution of $X$? Is it possible to express in terms of some ...
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1answer
158 views

Finding the distribution function of a random variable using CLT

Let $f_0$ and $f_1$ be two continuous probability density functions with means $\mu_0,\mu_1$ and variances $\sigma_0^2,\sigma_1^2$ on $\mathbb{R}$. Furthermore, let $l(y)=f_1(y)/f_0(y)$ be the ...
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2answers
98 views

Bayesian Inference/maximum Likelihood

I am rather struggling with the gist part d) of this question. Why would I wish to compare the MLE with the posterior mean?
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1answer
48 views

Let $(X,Y)$ be an absolute continuous R.V. Find $XY \mid Y = y$ and show $XY$ and $Y$ are independent. Also $XY \sim e(1)$.

Let $(X,Y)$ be an absolute continuous R.V with density $f_{X,Y}(x,y) = ye^{-y(x+1)}, \ x,y >0$. I've shown that $Y \sim e(1)$ and $X \mid Y = y$ density $x \mapsto ye^{-yx}$. However I must find ...
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152 views

What is the solution of the integral (product of two standard normal CDFs)?

I need to compute this kind of integral: where $b>0,d>0,a,c$ and $e$ are known constants and $\Phi$ is the CDF of the standard Normal distribution.
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1answer
120 views

Bivariate distribution of the sum and product of Gaussian distributed numbers

If $X$ and $Y$ are independent normally distributed random variables $$X,Y\sim\mathcal{N}(0,\sigma^2)$$ How are the sum and product, $X+Y$ and $XY$, co-distributed? You can write the moment ...
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1answer
45 views

Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$, i.e. $\forall \epsilon>0$, $\exists M_\epsilon>0$ such that $\mathbb{P}(X_T>M_\epsilon)<\epsilon$ $\forall T$. ...
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1answer
100 views

A point chosen at random from a disc

I've been working on this question and have managed to complete parts (i) through (iii) but am struggling with the last two parts. For (iv) I end up getting this when trying to find the distribution: ...
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1answer
27 views

Hypothesis testing using chi-square distribbution

Four players meet weekly and play eight hands of cards. Over a year, one of the payers finds that he has won x of the eight hands with frequency fx given in the following table: x 0 1 2 3 4 5 6 7 8 ...
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2answers
58 views

Why does $E[X]$ not equal the integral of $f(x)^2$

If $X$ is a random variable with the pdf $f(x)$ and $Y=g(X)$ how come $E[Y]$ is the integral of $g(x)f(x)$ but $E[X]$ is the intergral of $xf(x)$ ??
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1answer
65 views

the marginal pdf of a transformed variable from a joint distrubution

The questions tells us to let X and Y be random variables for which the joint p.d.f. is as follows: $$f(x,y)= \begin{cases} 2(x+y), & \text{for $0 \le\ y \le\ x \le\ 1$} \\ 0, & ...
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1answer
34 views

Probability distribution for the number of successes for $N$ distinct trials with distinct probabilities of success and failure

Imagine I have a process with $N$ distinct trials, $(t_1,t_2,t_3,t_4,...,t_N)$, where each trial $t_i$ has its own probability of success $p_i$ and probability of failure $q_i = (1-p_i)$. After ...
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1answer
58 views

stats - determine limiting distribution

Let $Y_{1} < Y_{2} < ... < Y_{n}$ be the order statistics of a random sample from a distribution with pdf $f(x) = e^{-x} , 0 < x < \infty$, zero elsewhere. Determine the limiting ...
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71 views

Finding the unknown constant of a PDF

$$f(x) = ke^{-x^2 / 2} $$ from negative infinity to positive infinity. I know that in order to find the constant you integrate the pdf with the limits and set it equal to 1. Unfortunately I haven't ...
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1answer
25 views

Moments of maximum of bivariate standard normal

Let $X,Y \sim N(0,0,1,1,\rho): f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}}$, and let $Z=max\{X,Y\}$. I'm looking for the first two moments of $Z$. I know it is ...
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Combining confidence intervals for sums of generic random variables

So my fiancee is a civil servant and asked me for help with the following problem. She has been given a collection of upper and lower bounds on expenditure for a collection of projects like: ...
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24 views

How to get from this probability formula to the one I need?

I'm working on a gambler's ruin problem where a player starts out with $i$ money, and 'winning' is when their total money reaches $N$ (ie they will keep playing until they reach N or run out of money, ...
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1answer
71 views

Find an unbiased estimator

Let $X$ be an r.v defined by $P(X=0)=p$ and $P(X=1)=1-p$. Find an unbiased estimator for $2p$. My solution: $E(X)=1-p$ so $2-2E(X)$ is unbiased. Is this correct?
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1answer
38 views

Weird question about probability density function

I'm assuming "actual" means the total probability of the PDF (the integral from $-\infty to \infty$) must be 1, so $$\int\limits_{-\infty}^{\infty} ke^{-0.1t}dt = 1$$ Wolfram Alpha seems to be ...
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1answer
472 views

Exponential Distribution calculation

I don't understand the following problem. ...