Questions tagged [random-variable]

A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).

Filter by
Sorted by
Tagged with
2
votes
2answers
159 views

Generation of random variables via composition and inversion

What are the main pros and cons of each method and when to use each one? Law [2007] mentions that: "Again, the reader is encouraged to develop the inverse-transform method for generating a ...
8
votes
2answers
133 views

Is it possible that marginally independent random variables are conditionally dependent?

Suppose that $X,Y$ and $Z$ are random variables. If $X$ is independent of $Z$ and $Y$ is independent of $Z$, is it possible that $X$ is dependent on $Z$ given $Y$ and $Y$ is dependent on $Z$ given $X$?...
4
votes
1answer
280 views

Is a “random variable” still a random variable if it is predictable?

If X in P(X) is considered a random variable because it varies among the occurrence the particular scenario which it may occur (such rolling a number on a die), can we still call X a “random variable” ...
-4
votes
2answers
52 views

Finding Probabilities of Normally Distributed Random Variables [closed]

A random variable $x$ is known to follow a normal distribution with mean $35$ and standard deviation $7$ Find the following probabilities: a. $P(x<25)$ b. $P(x<33)$ c. $P(x>42)$ d. $P(x>35)...
5
votes
2answers
107 views

Integral of cdf times pdf is a probability?

Let $X$ be a random variable with distribution function $F_X$. Consider $$P=\int_0^\infty (1-F_X(x))e^{-x}dx.$$ Because $1-F_X(x)$ is the probability of $X>x$ and $e^{-x}$ is the pdf of an ...
1
vote
0answers
35 views

What are the mean and variance of the square of a chi square?

Let $x$ be a random gaussian variable with mean=0 and sd=1, which is then squared (thus a chi-squared variable), so $y=x^2$. I understand that the expected value of $y^2$ is actually the variance of $...
0
votes
1answer
31 views

finding probability distribution of sum of 2 random variables

I have a probabiliy distribution $$p(x) = \begin{cases}e^{-x} & x\geq0\\ 0 & x<0\end{cases}$$ I need to find the probability distribution for $Z=X+Y$ where X and Y are from the above ...
3
votes
0answers
24 views

Truncated expectation of sum of independent random variables

Take three random variables $X$, $Y$, $Z$ s.t. $E[X]>0$, $E[Y|X]=0$, $Z = X+Y$. What can I say about $E[x| x> k]$ vs. $E[z| z>k]$ where $k>0$? Intuitively, the latter should be bigger ...
1
vote
2answers
20 views

Finding standard deviation of formula with three variables

Suppose we have the formula $Y = \frac{2WV^{2}}{\pi D^2}\hspace{1cm}$(1) where: w = weight variable, v = velocity variable, d = diameter variable we want to find $SD(Y)$ the solution which was ...
2
votes
2answers
61 views

Intuitively understanding $F_X(X)$ (r.v. as argument) [duplicate]

When considering some cdf $F_X(x)$ — e.g. from here — I’m having a hard time trying to understand what $F_X(X)$ really means. Expanding gives $P(X \leq X)$, which at first glance should always equal $...
2
votes
0answers
38 views

RV's with same mean/variance satisying Ohlin's lemma

I am trying to find two random variables X,Y with same mean and variance such that Ohlin's Lemma holds. That is, there exists some $x_0$ such that $F_X(x) \leq F_Y(x)$ for $x < x_0$ and $F_X(x) \...
2
votes
2answers
65 views

Why is the explanatory variable non-stochastic or fixed in repeated samples?

I am studying econometrics. I have been learning about deriving the variance for the OLS slope statistic in a simple linear regression model. Why is the explanatory variable considered to be non-...
2
votes
1answer
36 views

Find PDF of Z=X/(Y+c), c a constant and given independence of X and Y and the PDF of X and the PDF of Y

I want to find the PDF of $Z=X/(Y+c)$ where $c$ is a constant and $X,Y$ are two independent random variables. The PDFs of $X$ and $Y$ are supposed to be given. I would like to have a general form ...
0
votes
1answer
30 views

Z-scores and Probability

If $H$ is a normally distributed random variable with expected value 1.52 with standard deviation of 0.74. What is the probability that p(H=1.52)? Maybe I'm overthinking this but would you simply find ...
0
votes
0answers
33 views

Covariance matrix as a linear transformation

I am trying to understand the general relationship between the covariance between two random variables and linear transformations. For example, consider the figure here: https://en.wikipedia.org/wiki/...
1
vote
0answers
12 views

Good measures to quantify the effect of outliers

I have two random quantities linked to a set of N random variables. Increasing N The two random quantities should gradually become similar (meaning that the 2 distributions narrows around the same ...
1
vote
0answers
31 views

What is this probability distribution?

Thank you in advance for any suggestions or feedback. I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$. I am interested in finding the Wasserstein (...
0
votes
1answer
91 views

How does the sample variance change if you take subsets of $n$ observations from the original data?

Suppose $X$ is a continuous random variable for the weight of silver pennies. We then measure 338 pennies (in grams), leaving us with 338 observations. The observed mean weight is 15.722 grams and the ...
1
vote
1answer
23 views

Variance identities given $E(x)<\infty$

ok i got two identities i want to prove (true or false) $Var\left [ \left ( X-E(X) \right )\frac{1}{E(X)} \right ]=\frac{E(X^2)-E(X)^2}{E(X)^2}$ prove since $Var(aX)=a^2Var(X) $ , $Var(a+X)=Var(X) $...
0
votes
0answers
29 views

Mixed vs Fixed effects model

I'm currently weighing up the most appropriate of two models, one including individual as a random effect and the other discarding it: - ...
3
votes
1answer
27 views

Upper bound for absolute third central moment

Suppose $X\in \mathbb{R}$ is a random variable with expected value $\mathbb{E}X = \mu$. I ran across a proof which uses the inequality $$ \mathbb{E}[|X - \mu|^3] \leq 2^3 \mathbb{E}|X|^3. $$ Can ...
0
votes
0answers
31 views

Correlation and expected values

Consider two random variables, $x$ and $y$. Denote the correlation between them by $\rho$. Assume that $E[x]$ is also a function of some parameter $\pi$ and is increasing in $\pi$. So if we increase $\...
3
votes
3answers
43 views

Mean absolute difference for the gamma distribution

A wikipedia entry states that the mean absolute difference for the $\Gamma(k,\theta)$ distribution is $k\theta(4I_{0.5}(k+1,k)-2)$ where $I_z(x,y)$ is the regularized incomplete beta function, equal ...
5
votes
1answer
59 views

Can this problem really be solved using central limit theorem?

My friend had this question on a test: Let $\{X_n\}_{n \in N}$ be a sequence of independent random variables with the same normal distribution $N(0, 2n)$. Check for the convergence of a sequence $\{...
0
votes
0answers
18 views

What are the random variables that constitute i.i.d. samples?

Consider the following passage from Wooldridge (2010): "For much of this book we adopt a random sampling assumption. More precisely, we assume that (1) a population model has been specified and (...
1
vote
1answer
14 views

How to ascertain positive correlation between random variables?

Suppose we have random variables~$X,Y,Z$ that are related by the following: \begin{equation} X = Y + f(Z) \end{equation} for some function $f$. Under what conditions on the random variables and $f$ ...
3
votes
4answers
113 views

how to generate data from cdf which is not in closed form?

i am working on a distribution whose pdf and cdf is $$f(x,\alpha,\beta)=\frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta}}{(1+(\frac{x}{\alpha})^{\beta})^{2}}\frac{\sin(\frac{\pi}{\beta})}{(\frac{...
0
votes
0answers
15 views

Interactive plot of Region in Economic Model

So let's say I have the following framework. Let $s_1 \sim N(\mu_1,\sigma_1)$ and $s_2 \sim N(\mu_2,\sigma_2)$. Denote by $f_i$ and $F_i$ their pdf and pdf respectively. Define the following two ...
0
votes
0answers
15 views

A Sample is a a Single Data Point, or a Pool of Data Points?

This question has confused me a lot in statistics. I think in Statistics, a sample is a pool of data points from the PDF, rather than a single data point, am I correct? In everyday language if you ...
1
vote
1answer
27 views

exponential of expected value

If for a random vairable $X$ we have that : $$e^{\mathbb{E}[X]} = \mathbb{E}[{e^{X}}],$$ how can I show that $$X= c$$ almost sure, where $c$ is constant? Proof: Suppose that $$ \exp\left(\int_X f(x)d\...
0
votes
1answer
28 views

When I repeat an experiment 10 times, do I have 10 different random variables all of which are from that sample space?

I toss 2 dice to get their sum; now, this means I have a sample space from 2 to 12. Now, when I repeat this experiment 10 times, do I have 10 different random variables all of which are from that ...
3
votes
0answers
24 views

Expectation of uniform variates

Let $X_{1},X_{2},X_{3}$ be random variates from $U(0,1)$. It is required to compute $E(\frac{X_{1}+X_{2}}{X_{1}+X_{2} + X_{3}})$. Here is what I did.. $E(\frac{X_{1}+X_{2}}{X_{1}+X_{2} + X_{3}}) = E(1 ...
3
votes
0answers
18 views

Bounding values of a Dirichlet distribution

Consider $k$ random variables $X_1, X_2, \ldots, X_k$ such that $(X_1, X_2, \ldots, X_k)$ follow a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution. For a large enough $k$, I am trying to bound/find ...
3
votes
1answer
46 views

PDF of $X^2+2aXY+bY^2$

It is my first post on this forum. I am not a mathematician (so excuse me if I don't use the right vocabulary). I have two independent Normal random variables $X$ and $Y$: \begin{aligned} X&\sim N(...
5
votes
2answers
156 views

Distribution of the ratio of a Normal distribution divided by Lognormal distribution

I want to know the distribution (and the moments) of a variable, $Z = X/Y$, where $X\sim \mathcal{N}(\mu_{x}, \sigma^{2}_{x})$, and , $Y\sim \text{Lognormal}(\mu_{y},\sigma_{y})$? Hence, what I want ...
2
votes
2answers
25 views

Relationship between deterministc function of random variables

Given a discrete $P(X,Y,Z)$ let's call $\Omega$ the set of all deterministic functions $f: XYZ \rightarrow W$ and $\Omega'$ the set of all deterministic functions $f': XY \rightarrow V$. Is it correct ...
2
votes
0answers
40 views

Probability of averages

I have a random variable $Y$ and I am taking an independent sample of $n$ from this RV. I'll refer to this sample as $Y_n$, and I define the average of this sample as $\bar{Y}_n$. The maximum of this ...
1
vote
2answers
27 views

Identical Random Variables

I am reading the book "Probability - for the enthusiastic beginner" by David Morin. The book makes the following statement about Identical random variables Xi. " The sum X1 + X2 + X3 + ....
1
vote
0answers
34 views

Question about coin tosses and conditional probability

Case 1: Fair coin Consider a coin that is known ahead of time to have probability P=0.5 for heads or tails for a coin toss. Tom claims he can predict whether a toss will come up heads or not more ...
0
votes
1answer
18 views

Probability of discrete-random process

If $N_{\tau}$ is random process of number of items sold in $\tau$ minutes by probability below $$ \begin{align*} P_{N\tau}(n)=(5\tau)^n e^{-5τ}/n! \end{align*} $$ for $n=0,1,2,...$ Then imagine you ...
0
votes
0answers
29 views

How to compute the covariance matrix of a multivariate bernoulli distribution?

Considering this toy example: Let $x$ be a random variable $x \sim \mathcal{N}(\mu_x, \Sigma_x)$ Where $\mu_x \in \mathbb{R}^2$ is the mean vector and $\Sigma_x \in \mathbb{R}^{2 \times 2}$ is the ...
1
vote
1answer
17 views

Condition for zero or non-zero bias under transformation of random variable?

Suppose $\epsilon$ is a zero-mean univariate random variable noise with finite variance. I would like to find the condition on the function $f$ so that: $$ \exists x \in R \ \mbox{s.t:} \mathbb{E}[f(...
2
votes
1answer
52 views

Product of a Gaussian by a Beta random variable

I'm trying to find the distribution of a random variable $Z = X \cdot Y$, where $X \sim N(\mu,\sigma^2)$ and $Y \sim \text{Beta}(\alpha,\beta)$ with $\alpha$=1. I have tried with the convolution ...
0
votes
0answers
29 views

Linear mixed effect model: Small and unbalanced number of repeated measures per individual

I have a data set which contains 2 or 3 repeated measurements taken from 50 individuals over a 6 year period. The time between measurements is inconsistent. My goal is to estimate the effect age (...
0
votes
0answers
13 views

Population distributions/data generating functions in Bayesian Statistics

In many frequentist stats courses, random variables come from some distribution at the population level and as such we could say that $y=X \beta + \epsilon$ is the true function for something like ...
-1
votes
0answers
22 views

How can $X$ be a discrete random variable? [duplicate]

Suppose that the cumulative distribution function of discrete random variable $X$ is given by, $$F(x) = \begin{cases} 0 & \text{$x$ < 0 } \\[1.5ex] \dfrac{x}{4} & \text{$0 \leq x<1$}\\[...
1
vote
1answer
95 views

Mann-Whitney Normal Approximation process help

Let $X_{1}, X_{2}, ..., X_{n}$ is i.i.d sample from $X$ and $Y_{1}, Y_{2}, ..., Y_{m}$ is i.i.d sample from $Y$. And both samples are independent each other. Trying Mann-Whitney U-test then, $U =$ $\...
0
votes
0answers
8 views

Derivation of Pearson's Product Moment Correlation Coefficient's (PPMCC) distribution from bivariate normal variables?

I'm interested in reading through the derivation of the probability density function that PPMCC follows when it's input is a bivariate normal variables. Mathworld gives the following equalities: $$P(r)...
2
votes
2answers
45 views

Is this problem calculable only due to the parameter choices?

I am looking at a problem form Hogg, Tannis & Zimmerman (Ed. 10), and I am curious if the given problem is calculable (for an upper-level undergrad math/stats course) because of the choice of the ...
2
votes
1answer
54 views

Show $X_n \to 0$ in probability

I am asked to show : Let $X$ be a real-valued random variable on $(\Omega, F , P)$ and define $X_n(\omega) = nX(\omega)$ if $n<X(\omega)\le n+1$ and $0$ if else. Prove that $X_n \to 0$ in ...

1
2
3 4 5
40