0
votes
1answer
21 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
0
votes
0answers
6 views

Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
0
votes
1answer
16 views

Joint distribution and Integration

I was trying to prove a problem in my notes and now I need to whether prove or disprove the following claim: Assume $X,Y,W,Z$ are random variables defined on $(\Omega,\mathcal{F},P)$. If $(X,Y)$ and ...
0
votes
0answers
17 views

Formula for a Skewed Distribution

I'm trying to create a model of some data, using a skewed normal distribution. I have the following data: Mean Median Standard Deviation from the mean Standard Deviation from the median I've been ...
0
votes
1answer
39 views

Inferring symmetry of a distribution from its marginals

Let $X=[X_1,\ldots,X_n]$ be a continuous random vector of size $n$ with density function $f_X(x_1,\ldots,x_n)$. If all the marginals \begin{align*} \int \ldots \int f_X(x_1,\ldots,x_n)\, ...
-1
votes
1answer
30 views

$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

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

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...
1
vote
0answers
84 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 ,& ...
2
votes
1answer
78 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 ...
2
votes
0answers
12 views

Convergence of Quantiles moments.

QUESTION: Let $F$ be an absolutely continuous distribution function with density f, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
0
votes
1answer
41 views

$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find the MLE for $θ$.

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 ,& ...
0
votes
1answer
26 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) ...
1
vote
2answers
34 views

$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ 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 ,& ...
1
vote
1answer
28 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, ...
0
votes
2answers
23 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?
1
vote
0answers
27 views

Moments of stable random variables [closed]

I read in Handbook of Heavy Tailed Distributions in Finance that it is a 'well known fact' that: The $p$th absolute moment of a symmetric stable random variable (with index $\alpha \in (0,2) $) is ...
0
votes
0answers
50 views
+50

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 $$ ...
0
votes
1answer
32 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 ...
-2
votes
0answers
22 views

Help with random variable to found probabilty (PDF)

Stuck in this example to found (PDF) in many conditions
0
votes
0answers
30 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 ...
1
vote
1answer
39 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 ...
1
vote
1answer
35 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 ...
0
votes
0answers
14 views

Proof of “changes of sign” in one-dimensional random walk model [Feller's section 3.5, page 84]

Consider the one-dimensional random walk of a particle. We shall denote the individual steps by $X_1, X_2, \cdots$ with $X_i = \pm 1$ and the positions by $S_1, S_2, \cdots$ with $S_i = X_1 + X_2 + ...
3
votes
1answer
34 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 ...
1
vote
0answers
24 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), ...
1
vote
2answers
41 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 ...
0
votes
1answer
93 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 ...
1
vote
0answers
44 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 ...
0
votes
2answers
44 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}) = ...
1
vote
1answer
15 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 ...
2
votes
1answer
65 views

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

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 density function $ f $ is given by $$ f(y) = \begin{cases} ...
0
votes
0answers
24 views

How to represent?

You are a well-known hedge fund manager in Wall Street circles. One of your wealthy clients has $\$1$ million dollars to invest in XYZ stocks. Currently, XYZ stocks are trading at $\$2$ per share. You ...
1
vote
0answers
37 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 ...
1
vote
2answers
41 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 ...
0
votes
1answer
28 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 ...
3
votes
1answer
52 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 ...
0
votes
1answer
38 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 ...
1
vote
1answer
39 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 ...
0
votes
1answer
25 views

Let $(X,Y)$ be a random vector. Show $P(x,y) > 0$ implies $P(y) >0$ and $\sum_y P(x,y) = P(x)$ for $X = x, Y = y$ using the axioms of probability?

Let $(X,Y)$ be a random vector. How does one show $P(x,y) > 0$ implies $P(y) >0$ and $\sum_y P(x,y) = P(x)$ for $X = x, Y = y$ using the axioms of probability ? (In the continuous case the ...
2
votes
2answers
49 views

Compute the density of $Y=|X|$

When $X$ has the normal distribution $\mathcal N(\mu,\sigma^2)$ , compute the density of $Y=|X|$ I know ...
0
votes
1answer
29 views

Characteristic function of the Binomial distribution converges to that of the Poisson

Find conditions on $\lambda, n, p$, so that the characteristic function of the Binomial converges to that of the Poisson Binomial distribution is given as $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ ...
0
votes
1answer
26 views

if the CDF is non-invertible or does not have a closed form solution(e.g. Normal CDF), how can we generate random data from such a distribution?

Given the CDF of a distribution to generate random data from that distribution by using the inverse transformation of the CDF. Then if the CDF is non-invertible or does not have a closed form ...
1
vote
0answers
13 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
2
votes
2answers
33 views

Difference Uniform rv's

Let $U_{1}\sim U(0,1)$ be a standard uniform random variable. Is $U_{1}-U_{1}$ uniformly distributed? I've been trying to work this out as follows: Let $A,B$ be rv's $$P(A-B\leq ...
0
votes
2answers
54 views

Difficulty finding Expectation of a special function

I have a special function given as: $${\rm f}\left(r\right) ={1 \over \beta\lambda}\,2^{r/\beta} \exp\left({\left[2^{r/\beta} - 1\right]K \over \lambda}\right)$$ I should find the Expectation of ...
0
votes
1answer
20 views

Calculating probabilities of events

Was going through past previous exam questions and came across this one: A manufacturer of lie detectors is testing its newest design. It asks 300 people to lie deliberately and another 500 people ...
0
votes
1answer
47 views

For a poker hand, five cards are chosen from an ordinary deck of playing cards. How to find the probability to get the following hands

How would you find the of probabilities: (a) a hand with 1 heart, 2 diamonds, 2 clubs (b) a hand with no face cards (c) a hand with at least 3 queens
1
vote
1answer
31 views

Probability of 2 identical events

My professor said that probability of 2 identical events in a very short amount of time (dt converges to 0) is 0. However, I did not agree with him about this. Is there a proof for that assertion? ...
0
votes
0answers
33 views

$\mathrm{Pr}(X_1 + X_2 + \cdots + X_n \le n) .$ [duplicate]

Let $X_1,X_2,\ldots,X_n$ be a sequence of mutually independent random variables. For each $i$ with $1 \leq i \leq n$, we are given that the variable $X_i$ is equal to either $0$ or $n+1$, ...