Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the 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
70 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
37 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
423 views

Exponential Distribution calculation

I don't understand the following problem. ...
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1answer
35 views

Hypergeometric distribution excercise

There are 3 red and 4 blue balls in a box. Someone picks two balls from this box randomly. What is the probability that if we pick 2 balls from the remaining 5 balls, we pick exactly 2 red. I ...
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1answer
41 views

Bivariate normal distribution when $\rho$ is 0

What happens to the bivariate normal distribution when $\rho$ is 0?The bi-variate normal reduces to a simpler distribution, but what is it? and how do you calculate the cdf then? What I have tried: ...
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1answer
38 views

T statistic has t(n-1) distribution

I am trying to prove that $T_n=\frac{\bar{X}_n - \mu}{S/\sqrt{n}}\sim t_{n-1}$. One of the assumptions that seems to come up in proofs I saw of this is that ...
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2answers
34 views

CDF of Order Statistics

Why when finding the CDF of Yn and Zn do you find P(Y_n<=y) as opposed to P(Y_n<=y_n)? Similarly with Z_n.
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0answers
68 views

Question about computing the sample mean and variance values from a sample coming from a Weibull Distribution …

Let's suppose that I have a random sample x from a Weibul distribution with shape parameter k=1 and scale parameter λ=2... How am I supposed to compute the mean value of the sample ? Also what can I ...
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1answer
22 views

Test for Validity of Artificial Samples

I have a model that actually is learned from some observed samples. Then I use the model to generate several artificial data. My question is: Which test should I use to test if the data is of the ...
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0answers
39 views

Quantitatively comparing event trains of different lengths for Poissonness

I have a parameterized, effectively black box process that generates a series of events (simulated action potentials). Different parameter values often lead to different numbers of events. How can I ...
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1answer
61 views

Variance of a polynomial series from a uniform distribution

I intend to derive the variance of $Z$: $$Z \equiv \alpha_0+\alpha_1X+\alpha_2X^2+\dots + \alpha_MX^M = \sum_{m=0}^{M}\alpha_mX^m $$ for some $0 < M < \infty$ where each $\alpha_m \in ...
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0answers
43 views

Rolling a die $1000$ times, probability of getting $150\lt x \lt 167$ sixes, and probability of getting $x = 200$ sixes

If a fair die is rolled $1000$ times, what is the probability that a six is obtained between $150$ and $167$ times, what is the probability of getting 200 sixes? My logic: Noticing the probability of ...
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0answers
40 views

Gaussian independent, Mean, expectation, variance

Let $X$ and $Y$ be two independent Gaussian variables with zero mean and variance $\sigma^2$. Define:$$Z = |X-Y|.$$(a) Show that $\operatorname{E}[Z]= 2 \sigma / \sqrt{\pi}$. (b) Show that ...
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1answer
90 views

Poisson distribution with mean $\mu t$, arrival rate is $\mu = 15$ per minute

I have the following, with once again self-fabricated values, question: Let C(t) be the number of cats to arrive at a cat palace within $t (\geq 0)$ minutes. Suppose that C(t) has a poisson ...
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1answer
63 views

Infinite series containing binomial coefficients

I've encountered the following series: $$\sum_{t=1}^\infty {1 \over 2^{t}}\, {{\large t} \choose {\large{t + x \over 2}}}$$ Is this series even convergent? I'm really lacking knowledge on series ...
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1answer
185 views

Is the distribution of a product of M iid uniform random variables really Log Normal?

Conventional wisdom says yes or mostly. But consider the following simple derivation: Let $y = \prod_{i=1}^M x_i$ where $x_i\sim U(0,1)$. Then from independence, $E[y] = 2^{-M}$. Now, if we let ...
2
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1answer
62 views

Probability exercise Bernoulli. [closed]

Probability random signals. Im late I have no idea to start and this is for tomorrow. I was on training and have no break to do this work. I do this.You are an Internet savvy and enjoy watching video ...
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1answer
27 views

Bivariate Normal Manipulation

I do not understand the section of the solution highlighted.
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1answer
152 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
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0answers
864 views

Difficult integral for a marginal distribution

I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral: $$p(y)=\int_0^{\frac{1}{\sqrt{2 \pi }}} ...
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1answer
55 views

what value for $c$ yields the estimator for $σ^2$ with the smallest mean square error among all estimators of …

If $S'^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S^{'2}$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the ...
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1answer
79 views

Conditional Expectation Problem With Noise

I was given the following problem: let $X,N\sim \mathcal{N}(0,1)$ and let $A$ equal $1$ w.p $p$ and $0$ w.p $1-p$. Also, let $X,N,A$ be independent. Define $Y=AX+N$. Find $\mathbb{E}(X\mid Y)$. My ...
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3answers
106 views

Expected value of 2 Poisson distributions

Let $X$ and $Y$ be independet Poisson random variables with parameters $\lambda$ and $\mu$. I have to calculate $E((X+Y)^2)$ . What I did: $E[(X+Y)^2]=E[X^2]+E[Y^2]+2EXEY$ I know that ...
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1answer
217 views

If X and Y are equal almost surely, then they have the same distribution, but the reverse direction is not correct

Show that if two random variables X and Y are equal almost surely, then they have the same distribution. Show that the reverse direction is not correct. If $2$ r.v are equal a.s. can we write ...
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0answers
36 views

How to evaluate this expectation value?

How to evaluate this expected value: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$ where $\xi_i\overset{ind}{\sim} N(0,1), ...
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2answers
54 views

Let $Y,Z$ be idd. RV's, $\ Z \sim e(1), \ Y \sim R(0,1)$. Find $f_{X,Y}$ for $(X,Y)$.

Let $Y,Z$ be idd. RV's, $\ Z \sim e(1), \ Y \sim R(0,1), \ X = \frac Z Y$. ($R(0,1)$ denote the continuous uniform distribution) Compute $P(X>1)$: I have $P(X>1) = 1-P(X \le 1) = 1 - P(Z\le ...
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1answer
179 views

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval.

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval. In particular, show that the minimum length interval satisfies $f_{(n+3)}(a) = f_{(n+3)}(b)$, where ...
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0answers
108 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
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1answer
19 views

Exponential distribution of runners

Neil and Patrick go for runs whose lengths are independent and identically distributed exponential random varaibles with $\mu =35$ minutes. 1) What is the probability that the first to finish ...
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2answers
338 views

show that MSE$(\hat{\theta}) = E[(\hat{\theta} − θ)^2] = V(\hat{\theta}) + (B(\hat{\theta}))^2$.

Using the identity $(\hat{\theta} − θ) = [\hat{\theta} − E(\hat{\theta})] + [E(\hat{\theta}) − θ] = [\hat{\theta} − E(\hat{\theta})] + B(\hat{\theta})$, I need to show that MSE$(\hat{\theta}) = ...
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1answer
26 views

Marginal Density Question

I am faced with the following question, which I think is quite simple, but I can't put together for some reason. Given that $f(x,y)=(6/5)(x+y^2)$ for $0<x,y<1$, ($f(x,y)=0$ everywhere else), I ...
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1answer
164 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Problem: Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose probability density function $ f $ is given by ...
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3answers
36 views

Probability excersice

If $Z$ is a Gaussian random variable with mean $\mu_Z = 0$ and variance $\sigma^2_Z = 1$, and $Y$ is defined as: $$Y=a + bZ +cZ^2$$ for some constants $a, b, c$ show that the correlation ...
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1answer
56 views

This is a probability mass function problem [closed]

You are an Internet savvy and enjoy watching video clips of your favorite artists. You normally download video clips from the Web site http://www.coolvideos.com. The probability that you can ...
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1answer
28 views

Exchangeability of a Joint PDF

I'm wondering why the exchangeability of the bivariate normal pdf, allows me to immediately write down the distribution of Y2, having found that of Y1.
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1answer
35 views

Poisson distribution probability question [closed]

The typos on a page in magazine has a Poisson distribution. The probabilty of a page having typo(s) is 0.2. What is the probabilty that there are exactly 3 typos on a page? Thanks in advance!
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1answer
67 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$
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0answers
75 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
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2answers
230 views

throwing a dice repeatedly so that each side appear once. [duplicate]

Pratt is given a fair die. He repeatedly throw the die until he get at least each number (1 to 6). Define the random variable $X$ to be the total number of trials that pratt throws the die. or ...
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1answer
37 views

Equivalence, identity of random variables

Suppose I have $X \sim \text{Uniform}(0,1)$ and $Y \sim \text{Uniform}(0,1)$ As we all know $X+Y$ is a triangular distribution. What of $X+X$? Surely this is uniformly distributed on the interval ...
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1answer
67 views

How to compute transfomed pdf under non-injective function

I have a two random variables $x,y$ which are both (independently) distributed accordingly to the triangular distribution $x,y \sim Tri(-1,1,0)$ where I used the definition from Wikipedia. Now, I ...
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2answers
57 views

Geometric distribution: Collecting cute cats contained conscientiously.

I have just exposed myself to the geometric distribution, and have five fabricated questions based on the following information: You are collecting cats from baskets. A basket contains a cat with ...
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1answer
105 views

Function of a uniformly distributed continuous random variable

Basically, I'd like to add $n$ random vectors in a 2 dimensional space of unit length and of angle $\theta$ relative to a global axis. The probability density function of the angle $\theta$ is a ...
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1answer
59 views

Sum of probabilities or mean of probability

My question is about being confused about two way of approaching a problem, which in this case lead me to the same solution. One method is very verbose, the other one is fast and clean. Let's ...
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0answers
38 views

Distribution of an independent random variable given a Poisson distribution parameter [closed]

Let $X_1$ and $X_2$ be two independent random variables. Let $X_1$ and $Y=X_1+X_2$ have Poisson distributions with means $\mu_1$ and $\mu > \mu_1$, respectively. Find the distribution of $X_2$. I ...
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1answer
61 views

How do I find $\theta$ with bootstrap?

I have two vectors of known values $x$ and $y$. And the relationship between them is $y=\sin(\theta \cdot x)+\epsilon$, $\epsilon \sim N(0,1) $ . The question is how do I estimate $\theta$ with ...
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1answer
125 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
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1answer
55 views

Question about probability of order statistics

I'm given two exponential random variables $x_1$, $x_2$ that both have a mean of $1$ (so $\lambda = 1$ for both r.v.s, presumably), and asked to solve for the probability: $$P(X_{(2)} \gt 3*X_{(1)})$$ ...