# 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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### Generating random variates from the following pdf

I'm working through some example of probability distributions and I'm struggling to derive the formula for the following pdf $f(x) = \frac{1}{0.02}e^{-\left\lvert x \right\rvert/0.01}$ My undersanding ...
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### Problem calculating expectation using law of total expectation

I'm confusing myself with conditional expectation and could really use your help! I am trying to calculate an expectation that arises in the context of doing variational inference. However, the ...
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### Independence of variables between X and Y^X

If X and Y are independent, then are X and Y^X independent? Does the realisation of X have to be the same as the X in the power of Y? I think this question sounds silly but I'm trying to clear a major ...
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### Need help understanding how only variable A can be correlated to the absolute value of A-B

I'm currently working with the dataset of a study I'm conducting. The data is comprised of serially drawn samples from patients where we've measured the cell counts of those samples and compared them ...
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### Dependent and not identically distributed random variables

I am trying to deepen my knowledge in probability and I am having hard times to understand dependent and not identically distributed random variables. Can someone maybe provide me a real world example ...
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### Generate Uniform Random Variates with Constant Norm [duplicate]

How can one generate $k$ uniform random variates centered at zero, $X_1, X_2, ..., X_k$, given a constant Euclidean norm, $c =\sqrt{X_1^2+X_2^2+...X_k^2}$?
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### What is the entropy of a riskless random variable?

Variance and standard deviation are often used as proxies for risk and volatility. I make the analogy to information theory as follows, correct if it's wrong: a random variable $x\in \mathbb{R}$ that ...
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### Correlation is a symmetric measure, but scatter plot matrix shows asymmetric dependence

The correlation matrix demonstrates that correlation is a symmetric measure: $\rho(X,Y) = \rho(Y,X)$ since the lower off-diagonals are mirror images of the upper off-diagonals. The scatterplot matrix ...
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### Simulating realizations of joint Bernoulli distribution

Let $X$ and $Y$ be Bernoulli random variables with success probability $p$ and $q$ respectively, i.e., \begin{align*} X = \begin{cases} 1 & \text{with probability $p$} \\ 0 & \text{with ...
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### Can I see i.i.d. variables as just one?

I'm trying to understand the part of this book, page 276 which explains about sample mean: In statistical inference, a central problem is how to use data to estimate unknown parameters of a ...
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### Converge of Scaled Bernoulli Random Process

Suppose a random sequence is defined by $X_n := n B_n$, where $B_n$ is a Bernoulli sequence such that $\mathbb{P}(B_n = 1) = 1/n$. I am interested in the convergence properties of this random process ...
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### What is the mean and variance for this Normal distrbution? [duplicate]

Suppose, we have $X_1,X_2,$ and $X_3$ i.i.d random variables (RV's) from $N(\mu,\sigma^2)$. Let $Y=\frac{X_1-X_2}{2}$, and $W=X_3-Y$. Then is $W\sim N(0,2\sigma^2)$ or $W\sim N(0,\sigma^2)$?
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### Derivation of skewness and kurtosis algebra of random variables

In algebra of random variables, the symbolic rule for computing variance of random variable $X\in\mathbb{R}^{n\times p}$ multiplied by a coefficent vector, $a\in\mathbb{R}^p$, is \text{Var}(X\cdot a)...
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### Confusion on independent variables and prediction

Suppose $X_{i}$ is an i.i.d random variable for $i = 1, 2, \dots, n$. Since $X_{i}$ and $X_{j}$ are independent for all $i \neq j$, the mutual information $I(X_{i}, X_{j}) = 0$. If $X_{i}$ has ...
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### Expected vallue calculation of i.i.d. random variables

Suppose $X_1,X_2,\ldots,X_n$ are a sequence of i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Define the sample mean $\bar{X} := \frac{1}{n} \sum_{i=1}^{n} X_i$, which we know is an ...
Let $X$ be a continuous random variable with cdf $F_{X}$ and pdf $f_{X}$, median $m≡F^{−1}X(\frac{1}{2})$, and $f_{X}(m)>0$. It is a well known result that $m= arg min_{b}E[|X−b|]$. Does this ...