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).

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Calculating expected value

Came across an interesting problem. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it to the ...
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probability that the players will exchange their initially drawn number

Consider the following two-player game. The players simultaneously draw one sample each from a continuous random variable X, which follows $Uniform\ [0, 100]$. After observing the value of her own ...
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Correlation and Max and Min of Two Exponentials [duplicate]

Came across an interesting problem: Let X and Y be independent random variables such that both X and Y ∼ Exp(1). Define L = min(X,Y) and H = max(X,Y). What is ρ(L, H)? Why must the correlation should ...
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Minimum and Maximum of Two Exponentials [closed]

I'm confused by how to figure this out. So far, I've defined D to be a random variable tracking the amount of time that passes between the first and second events, so that D = H - L. While deriving ...
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Studying extreme value r.v. $X=\max_i (c_i+X_i)$ where $c_i$ are constants and $X_i$ are i.i.d. r.v

Let $X_1,X_2,...,X_n$ be independently and identically distributed random variables according to a distribution $F$. There are constants: $c_1,c_2,...,c_n$. Define a new random variable $X=\max_i(X_i+...
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What is the conditional expectation of two random Poisson variables?

Say I have two random variables $Z^{p}_{i} \mid Y^{p}_{i} \sim \text{Poisson} \left(t^{p}_{i}Y^{p}_{i}\right)$ and $Z^{q}_{i} \mid Y^{q}_{i} \sim \text{Poisson} \left(t^{q}_{i}Y^{q}_{i}\right)$ for ...
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Explanation of $X: (\Omega,\mathscr A,P)→(\mathbb R , \mathscr B',\mu)$ [duplicate]

A random variable $X$ is a function with a finite value so that: $X: (\Omega,\mathscr A,P)→(\mathbb R , \mathscr B',\mu)$ I've seen this line but I don't know how to understand it. Can you please ...
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Visualization of $L(X)$

Let $L^2_+$ the set of all $2$-dimensional nonnegative random vectors $X = (X_1, X_2)^⊤$ with finite and positive marginal expectations, and let $Ψ^{(2)}$ the class of all measurable functions from $\...
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Equivalent definitions of $ L_2$ convergence?

I have been reading up on the convergence of random variables, and I have come across two commonly given definitions of $ L_2 $ convergence: $ \|X_n-X\|_{L_2} \to 0:$ $(1):\left(E|X_n - X|^2 \right)^{...
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A question about linear inference in random Fourier feature kernels

In Ali Rahimi's and Ben Recht's paper "Random Features for Large-Scale Kernel Machines," there is a line near the bottom of the introduction which I can not reason about... In addition to ...
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If $X_1 \sim \text{binom}(p_1,n_1)$ and $X_2 \sim \text{binom}(p_2,n_2)$, how to prove that the MLE of $p = p_1 - p_2$ is $\hat{p}_1 - \hat{p}_2$?

Suppose $X_1 \sim \text{binom}(p_1,n_1)$ and $X_2 \sim \text{binom}(p_2,n_2)$, where $X_1$ and $X_2$ are independent, and let $p = p_1 - p_2$. How can I prove that $\hat{p} = \hat{p}_1 - \hat{p}_2$? (...
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Exponential random variable X with a uniform random variable as its parameter

$$X\ \sim Exp(U) ~ and\ U\ \sim U(0,1) $$ The question asked for the value of $ P(X\geqslant 1)$ I saw the solution and it went like this: $$P(X\geqslant 1) = E[P(X\geqslant 1)|U] = E[e^{-u}] = \int_{...
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P(XY<1/2) , X and Y uniform independent RV [-1,1]

I think the probability should be equal to shaded area shown in the figure divided by 4, which is equal to (3+ln 2)/4. Is this correct, because my answer is not matching My solution: I find the area ...
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Variance of the sum of multiple random number generators

Let's assume I have "n" random number generators, each one has a different variance value, but has the same mean value, zero. If I generate "n" random numbers with these generators,...
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Probability that a Random Variable is Greater Than Another

Say I have two independent random variables - one is drawn from a uniform distribution on [0, 50] while the other is drawn from a uniform distribution on [0, 100]. How would I calculate the ...
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Distribution of a function of Normal and Uniform random variables

consider the distribution of $$Y=\sqrt{(A \sum_{i=1}^{k}\cos \theta_i+\Re{\{ N \}})^{2}+(A \sum_{i=1}^{k}\sin \theta_i+\Im\{N\})^{2}}$$ where $A>0$, $N \sim \mathcal{CN}(0,2\sigma^2)$ and $\Theta_i ...
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Estimate variable based on correlated random generated variable

I have two random variables $X$ (price of a commodity) and $Y$ (default rate of that commodity), that are correlated through $\rho$. $X$ follows a log-normal distribution and $Y$ has not enough ...
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Estimating the Probability in a statistical analysis

I am carrying out a statistical analysis where I run the simulation (Matlab) 5000 times, to get 5000 results. The objective is to estimate the probability of having a result that is less than or equal ...
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Computing Gini coefficient for a 2 parameters density function

I have a random variable $X$ defined by the following the density function, \begin{equation} f_{\theta_1, \theta_2}(x) = \begin{cases} \frac{\theta_1 \theta_2^{\theta_1}}{x^{\theta_1 + 1}}, &...
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What is the pdf of the multiplication of two normal random variables? [duplicate]

I want to know the pdf of the multiplication of two normal random variables (may or may not have the same mu and sigma, may or may not be correlated). ...
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Covariance of a sample statistic for two independent bivariate random variables

I have a somewhat convoluted question here. Suppose I have paired random variables $X$ and $Y$. That is, when I draw samples, I get one instance of $X$ and an associated instance of $Y$. Then I can ...
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Test statistics to show randomness on stock returns vs. some economic event

I have a data set which contains economic events (per individual equity security), I want to do some analysis on where these events have any impact on individual equity stock returns. My initial data ...
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covariance of lognormal random variables

I am trying to find the variance of b*log(x+y) - log(x), where x and y are independent and identically distributed lognormal random variables, the range for log(x) and log(y) is negative infinity to ...
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What is the term for $F(x)/[1-F(x)]$ where $F(x)$ is CDF

In a research project, I developed a result which consists of a condition on $F(x)/[1-F(x)]$, where $F()$ is the CDF of the random variable $X$. What is the term for $F(x)/[1-F(x)]$? Does it have any ...
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Approximated division of a sample (in the infinite divisibility sense)

Suppose i have a sample $x_1,...x_n$ from a random variable $X$ that is supposed to be infinitely divisible. Since $X$ is $2$-divisible, there exists $Y$ and $Z$, independant and with the same ...
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Density function of nonlinear combination of normal random variables

Say we have two random variables $A,B \sim \mathcal{N}(0,1)$ and they form the following combination $$ X = A^2 + B^2 - \frac{A^2 B^2}{A^2 + B^2}. $$ Is there any way to obtain the probability ...
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Does the marginal distribution assume independence? (i.e. sampling without replacement)

Let $X_1, X_2, ... , X_n$ be a sample drawn without replacement from a finite population. $X_1$ may be the random variable - weight of the first person; $X_2$ may be the random variable - weight of ...
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Mistake in Casella & Berger on page 207?

Page 28: A note on notation: Random variables will always be denoted with uppercase letters and the realized values of the variable (or its range) will be denoted by the corresponding lowercase ...
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distribution of normalized random weights

TL;DR: Given a finite series of positive and iid random variates/weights $w_i$ of known distribution, $(w_1, ... w_n)$, what is the distribution (or at least the exp.val. and variance) of the ...
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How do I fit a nonlinear mixed effects model with multiple observations from the same sample units using nlmer()?

I am trying to fit a nonlinear mixed-effects model with lme4. However, I am not sure how to adequately account for my sampling design. The model is supposed to ...
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3 votes
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Interval for the expected value

X is a Random Variable that can take values only in $[0,10]$. Suppose $P[X>5] \leq \frac{2}{5}$ and $P[X<1]\leq \frac{1}{2}$. Then what is the interval for the $E[X]$? Answer : $E[X]\geq0.5$ and ...
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Covariance of ratio of dependent variables?

I am trying to use the Delta method (Please have a look at this link) to compute the covariances between the ratios of random dependent variables. I have 7 dependent variables $A_i$, $i\in\{1,2,3,4,5,...
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Dependend or independend random variables?

i have two sets of values for soilmoisture, A and B. One set, A, is coming from a reanalysis model, thus it is simulated based on measurements of 2m temperatures given by observation stations. The ...
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probability mass function of sum of two dependent discrete random variables

Let $R_1$ and $R_2$ denote two real-valued discrete random variables. The probability convolution is the probability distribution for the sum $R_3:=R_1+R_2$ when $R_1$ and $R_2 $ are independent. For ...
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Var and Expected Shortfall

I am struggling to find an example which has 2 random variables (say L1 and L2) with same VaR but different Expected Shortfall.
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Does increasing the variance increase the value of a function?

Let $V=\sum_{i=1}^{k} a_iX_i$ where $X_i's$ are IID $\sim Bern(q)$ and $V$ with $\sum a_i=k$. Note that $a_i$'s are non-negative integers. I have a function $f$ as given below : $$ f= \max_{q} h\...
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2 answers
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Formula for the variance of the product of two random variables [duplicate]

Let's say I have two random variables $X$ and $Y$. Can I write that: $$VAR \left[XY\right] = \left(E\left[X\right]\right)^2 VAR \left[Y\right] + \left(E\left[Y\right]\right)^2 VAR \left[X\right] + 2 \...
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3 votes
1 answer
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Why is each observation in a sample considered a random variable in linear regression?

I have the following excerpt in my statistics textbook: I am confused by the sentence: "Another way statisticians treat this model is that, assume $X_1...X_n$ are random variables, we make ...
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6 votes
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How to find the PMF of a weighted sum of IID Bernoulli random variables with constant sum of weights

Let $\{X_1,X_2,\ldots X_k\}$ denote a set of $k$ IID $Bern(p)$ random variables. Also, I have a set of $k$ non-negative integer weights denoted by $\{a_1,a_2,\ldots a_k\}$ such that $\sum_i {a_i}=k$. ...
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Variance of the sum of N correlated random variables with equal variance [duplicate]

According to this Wikipedia article, In general, the variance of the sum of $n$ variables is the sum of their covariances. So if the variables have equal variance ${\sigma}^2$ and the average ...
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Integral of conditional expectation over an event B

Could someone help me understand this equality? Let $\xi$ be a random variable. $\int_B(\frac{1}{P(B)}\int_B\xi dP)dP=\int_B \xi dP$ for any event $B$. How do we go from the integral over an event $B$ ...
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1 vote
1 answer
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Distribution Function from Density Function

I'm guessing there was an error in a Probability and Statistics exam I have recently taken. Let $X$ be a random continuous variable and $f$ a function defined as follows: $ f(x)=\left\{\begin{matrix} ...
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Hypothesis testing with difference of means of two random variables [duplicate]

In this problem, I thought in an answer that I cannot figure out why is giving me an incorrect result. The problem is: Given: $X$ and $Y$ as independent Bernoulli Distributed random variables and $...
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1 answer
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Rewriting expressions involving Bernoulli random variables

my problem is the following. Consider only Bernoulli random variables $X_1,\dots, X_n$ where $P(X_i = T) = p_i$ ($T$ stands for true, success). All the r.v.s are independent. Starting from the ...
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2 votes
1 answer
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Implied expressions with same probability

Given two variables $a$ and $b$ and two intervals $I$ and $J$, is the following affirmation true? $$ \Pr[a\in I]\geq p\ \land\ (a\in I\implies b\in J) \implies \Pr[b\in J]\geq p $$ Of course, $a\in I$ ...
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5 votes
1 answer
627 views

Show that, for any real numbers a and b such that m ≤ a ≤ b or m ≥ a ≥ b, E|Y − a| ≤ E|Y − b| ,where Y be a random variable with finite expectation

Let $Y$ be a random variable with finite expectation, and $m$ be a median of $Y;$ i.e., $P(Y \le m) \ge 1/2$ and $P(Y \ge m) \ge 1/2.$ Show that, for any real numbers $a$ and $b$ such that $m\le a \le ...
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What is $E[X]$ and $\text{Var}(X)$ if $X$ follows $Pois(T^2)$ and $T$ follows Exponential distribution [duplicate]

I'm new to this community. I have problem in finding expected value and variance of R.V.s that are composed of other R.V.s following other distributions. Suppose $X \sim Pois(T^2)$ where $T \sim Exp(\...
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1 vote
1 answer
36 views

conditional expectation of random vectors

For a random vector $X=(X_1, ..., X_n)^\intercal$ the expectation value can be written as $\mathbb{E}[X] = (\mathbb{E}[X_1], ..., \mathbb{E}[X_n])^\intercal$ according to equation 2 in https://en....
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Expected distance under Gaussian noise

Summary I'm working on a tracking problem, where I'm trying to estimate the position of an object that moves in on plane. In my simulator, at each sampling step I generate a measurement that is given ...
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2 votes
1 answer
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Expected number of points in a subsurface of a rectangle

There is a rectangle $S$, and $n$ points are uniformly distributed inside it. If we select an area $A$ inside the rectangle, what is the expected number of points inside the $A$? I think it seems to ...
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