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How to find R's domain when using the inverse transformation method to create random-variate generation?

The cumulative distribution function (CDF) of a random variable X is a function F(x) that gives the probability that the random variable X is less than or equal to x. In other words, F(x) = P(X ≤ x). ...
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2 votes

Prove that $0\leq (\mathbb{E}[X])\leq (\mathbb{E}[X^2])^{\frac{1}{2}}\leq (\mathbb{E}[X^3])^{\frac{1}{3}}\leq ...$

This can be proved directly using Jensen's Inequality: Let $k$ be a positive integer and $X$ a nonnegative real-valued random variable. We show the inequality $\mathbb{E}[X^{k}]^{\frac{1}{k}}$ $\...
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1 vote
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An integer valued and $\sigma$-finite measure without atoms is costantly zero?

Let $\mu$ be a nonzero, integer valued measure on a complete separable metric space $S$ with metric $\rho$. We will show there exists $x \in S$ such that $\mu\{x\} >0$. Let $D$ be a ...
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1 vote
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$\varepsilon_{ij} \in \mathcal{N}(0,\sigma^2)$. What can we say about the third moment?

Hint: The function $$x\mapsto \frac{1}{\sqrt{2\pi}\sigma }x^{2n+1}e^{-\frac{x^2}{2\sigma ^2}},$$ is odd for all $n$.
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Conditional probability when X is uniform on [-1, 1]

The basic idea of the conditional probability $P(.|B)$ is to set the probability measure to zero outside B and to rescale it inside B so that $P(B|B) = 1$ So we have $P(B|B)=1$ and $P(B^c|B)=0$ (&...
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The distribution of the sum of sub-exponential random variables

Yes. We use the definition that $X$ is sub-exponential if there exist $\nu, \alpha > 0$ such that, for all $|\lambda| \leq 1/\alpha$, $$ \mathbf{E} \exp\{\lambda(X - \mathbf{E} X)\} \leq e^{\frac{1}...
3 votes
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Doob's Optional Stopping Theorem to find probabilities of stopping times

We find a martingale that will make things work out. Define $S_n=X_1+...+X_n$, we define $(c^{S_n})_{n \in \mathbb{N}}$ and find $c$ which makes the process a martingale; since the jumps are bounded, $...
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Expectation for Jointly Distributed Random Variables

It looks like you are finding the expectation of $XY$. See this link for formulas: discrete probability formulas In this case, $$E(X)=3a+1b+2c$$ $$E(Y)=2b+1c+3d$$
1 vote
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Does the integer-part of a continuous random variable still admit a density?

No, $Y$ does not admit a density with respect to Lebesgue measure. The support of $Y$ is $\{\frac{0}k,\frac1k,\dots,\frac kk\}$, which is a Lebesgue null set. If $Y$ had a density, then $\int_{S} f_Y(...
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1 vote

The definition of probability mass function. Does the random variable has to be discrete?

The notation $P(X=x)$ refers to the probability that $X$ is exactly $x$. For a continuous r.v., we consider the probability of $X$ within a range. The probability of $X$ equal to a particular value is ...
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Function of simple random walk is a martingale

Integrability comes from the fact that $|S_n| \le n$. In the case that $p \ge q$, we have $\mathbb{E}[|Z_n|] \le \left(\frac qp\right)^{-n} < \infty$ for all $n$. In the case that $q \ge p$, we ...
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1 vote
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Show a closer correlation between random variables

I would say that the claim that, for any weighting factors $w_1, \dots, w_n \in \mathbb{R}$, the weighted variable $Y = \sum_i w_iY_i$ is more closely related to $X$ than $Z = \sum_i w_iZ_i$ given ...
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CDF of standard norm dist random variable by division?

Considering that you said this was an exercise, I won't be providing the direct calculations. Question 1: So, if you look at the definiton of the Cauchy distribution, its pdf is given by $f_X(x) = \...
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Product of two stationary random processes

Regarding part 1, here is a discrete-time counterexample: Let $X_k$ be i.i.d. $\pm 1$ valued variables of mean zero, and let $\{W_k\}{k \in \mathbb Z}$ be an independent copy of $\{X_k\}_{k \in \...
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1 vote
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function of measurable random variable is measurable

It easily follows from the definition of a measurable map that the composition of two measurable maps is itself measurable. So any measurable map of $X$ would be measurable with respect to $\mathcal F$...
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$X$ is $\sigma(Y_{1}, ..., Y_{n})$-measurable iff $X = H(Y_{1}, ..., Y_{n})$ for some $H$ measurable

Here is an answer for the case where $X$ is a real random variable, and $Y$ a random variable taking values in any measurable space. In your situation, set $Y = (Y_1,\ldots,Y_n)$. If $X = 1_A$ for ...
1 vote

Consider a continuous random variable X. Find some value c such that $ P(X>c) < 0.01 $.

I followed at @David Mitra's advice by not expanding the integrand when taking the anti-derivative. After some mistakes with not updating the bounds I think I got the answer. $$ F_X(c) = \frac{3}{250}...
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How bad are 11 dry years in a row?

The easiest excel-friendly way to show the trend is to do a polynomial interpolation, a more sophisticated version of this would be to use a digital filter. However, apart from the approach @...
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Expected value of given random variables

Let $R_1, R_2, R_3$ denote the result of the first, second, and third roll, respectively. Then $X=\sum_{i=1}^3 R_i$ and $Y=\sum_{i=1}^3 R_i^2$. Then by linearity of expectation, $\mathbb{E}(X)=\sum_{i=...
1 vote
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Unif affine function for RV U Unif distributed

No, that function doesn't fill the bill. The function $h(u) = (b-a)u + a$ will do. Indeed, let $X = h(U)$ and note that $h$ is bijective, with inverse $h^{-1}(x) = \frac{x-a}{b-a}$. Thus, we can ...
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What is the value "a" here, and what does it represent?

$a$ is the $y$-intercept. It is the predicted value of $y$ when $x=0$, that is the GPA when the IQ is $0$.
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Why probability distribution is defined on event space and not on sample space?

The probability space is the source of all randomness. If there are multiple random variables defined on the same sample space, then any probability distribution on the sample space determines the ...
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There are two sets of random variables - iid (uniformly) on a circle - what is pdf of minimum distance between 2 points from different sets?

Notation. The unit circle can be denoted by $T$ for torus. The $n$-fold Cartesian product will be written as $T_n$. An interval $B\subset T$ is considered to be a one-dimensional ball whose center ...
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Why probability distribution is defined on event space and not on sample space?

For any continuous R.V., the value can be a decimals. For example, there are infinite amount of deciamls between 0 and 1, e.g. it can be 0.9 or 0.99999999 or 0.99999999999999. So the $P[X=x]$ for any ...
5 votes

Why probability distribution is defined on event space and not on sample space?

Short answer (leaving out all the technical stuff). For continuous probability the probability of a single point is $0$ and you can't get the probability of an event (say an interval) by summing the ...
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X and Y are independent, binomial random variables with different probabilities. How would I find the conditional probability of their sum?

More generally assume that $X\sim\mathsf{Bin}(n,p)$ and $Y\sim\mathsf{Bin}(m,q)$ are independent, and $Z=X+Y$. For $0\leqslant i\leqslant k\leqslant n+m$ we would have by definition of conditional ...
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Projections of functions of random variables

Your question contains very little information (about what $g$ is,... ),​so it quite hard for us to give you a satisfying answer. You might be interested in Orlicz spaces, and thus Orlicz norms. They ...
1 vote
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Find probability density function for random variable

For any Borel $A \subset \mathbb{R}$, $$P(h(X) \in A) = P(X \in h^{-1}(A)) = \int_{h^{-1}(A)}f(x)\,dx = \int_{A}f(h^{-1}(y))\frac{1}{|h'(h^{-1}(y)|}\,dy.$$ Hence $$f_{h(X)}(y) = f_{X}(x)\frac{1}{|h'(x)...
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Characteristic function of random variable

First note that, since $E(X)=0$, $\mathrm{var}(X) = E(X^2) -E(X)^2 = E(X^2)$. Now you can use your theorem with $n=2$ as you are doing. To obtain the final assertion, first note that $E(X^0)=E(1)=1$ ...
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2 votes

How to find the $CDF$ of random variable $Z = max/min(X,Y)$?

Hints: You want $P(Z_1 < z_1$ and $Z_2 < z_2)$ '$Z_1 < z_1$' is the same as '$X < z_1$ and $Y < z_1$' because maximum(X,Y) less than something is equivalent to saying both X and Y are ...
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4 votes
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Is it true that $\sigma(X_1,X_1+X_2)=\sigma(X_1,X_2)$?

Elaborating on my comment: For a measurable function $f$, $$\sigma(f(X)) \subseteq \sigma(X),$$ since a set $\{\omega : f(X(\omega)) \in B\}$ can be expressed as $\{\omega : X(\omega) \in f^{-1}(B)\}$....
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2 votes
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Joint Distribution of dependent or independent random variables

In case there is a one-to-one function from $X$ to $Y$ (like $Y=4X+3$), then knowing $Y$ directly tells you the value of $X$ and vice versa. This means that probabilities like $$P(X=4\mid Y=19), \quad ...
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Joint Distribution of dependent or independent random variables

Taking into account the definition of independence, I.e, $P(X=x, Y=y) = P( =x)P(Y=y)$ we see that the two are not independent. Consider n = 35. $P(x = 2) = 1/35$. $P(Y= 83) = 1/35$. $P(X= 1|Y = 83) = ...
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Substitute in what is known in conditional expectation

A sufficient condition for (1) to be true is that there exists a regular conditional distribution for $X$ given $\mathcal{G}$. This is true, for instance, if $S$ is a standard Borel space. A regular ...
0 votes

Bounded variance for Lipschitz function of random variable

You can use similar reasoning to that of @Nanayajitzuki to improve their suggested bound by a factor of 2. Specifically, take $X, Y$ i.i.d. Then, $X$ and $Y$ are uncorrelated being independent, i.e., $...
1 vote
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Show that $X_n\overset{a.s}{\to}X$ and $M_n\overset{P}{\to}\infty$ implies $X_{M_n}\overset{P}{\to}X$

Convergence of $(U_n)$ to $U$ in probability is equivalent to the fact that every subsequence of $(U_n)$ has a further subsequence converging a.s. to $U$. Your statement follows easily from this.
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Bivariate Continuous Random Variable - Double Integral Calculation

Graphing the region defined by this function yields a sort of filled triangle bounded by the lines y1=y2, and y2=2−y1 enclosed by the y-axis. So that is: $y_1$ is less than the minimum of those two ...
1 vote

Exact expression of variance of gaussian quadratic form

Since $\vec{x}$ is a standard gaussian random vector, and if $\mathbf{A}$ is symmetric, then from 1 the first moment of $\vec{x}^\prime\mathbf{A}\vec{x}$ is $$\mathbb{E}[\vec{x}^\prime\mathbf{A}\vec{x}...
2 votes
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Prove that $T$ is a stopping time

A key thing here is that $\sup \emptyset = -\infty$. First, $\{T> n\}=\bigcap_{k\leq n}\{X_k \notin B\}\in \mathscr{F}_n,\,\forall n$ so $\{T\leq n\}=\{T>n\}^c \in \mathscr{F}_n,\,\forall n$. ...
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Random Walk Stopping Time Calculations

Define the function $ f_{p} = f : \mathbb{R} \to (0, \infty) $ by \begin{equation*} f_p (\lambda) = p \exp (-\lambda) + q \exp( \lambda ) . \end{equation*} We have \begin{equation*} f^{\prime}(\lambda)...
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Find the bounds of a random variable $A$ from $X(t) = acos(At + V)$ where $V \sim Uniform[0, 2\pi]$

Hint: $\cos x$ is a periodic function, and you can write a general A as $$A = 2 k \pi+ A_{[0,2 \pi]}$$ Where $A_{[0,2 \pi]} = A \mod 2\pi$ and $k \in \mathbb{Z}$.
2 votes
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For any random variable X, the propability that X>E(X) is 1/2?

The claim in the title is not true. For a counterexample, let $X$ be a Bernoulli random variable with success probability $p$, where $p \in[0,1) \setminus\{1/2\}$. Then $\mathbb{E}(X) = 0(1-p)+1p = ...
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Multiply a random vector by an orthogonal matrix. The result has the same distribution.

This follows from the more general result: if $X \sim N(0, I_n)$ is a multivariate normal vector in $\mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ is a fixed matrix (assumed to be invertible/full ...
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1 vote
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Free Poisson Distribution

For any given $n$ we have $$ \mathbb P\left(\sum_{j=1}^n Z_{j,n}=k\right) = \binom nk \left(\frac\lambda n\right)^k\left(1-\frac\lambda n\right)^{n-k}. $$ As $n\to\infty$ the above limit is $e^{-\...
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Different ways of calculating a bivariate probability

It's easy to get trapped like this: $$ \begin{align} P(Y > 2X) &= E(P(Y>2X|X)) \\ &\color{red}{\neq} \int_{x=0}^\color{red}{1} P(Y>2x)f_X(x)dx = \int_0^1 (1-2x)2xdx\\ &= 2\int_0^1 ...
2 votes

Expectation related to geometric random variable

They are actually equal: $\sum_{k=1}^\infty k\mathbb P[Z= k] = \sum_{k=1}^\infty \mathbb P[Z\geq k] $ Consider this sum: $ + \; P(1) \\ +P(2) + P(2) \\ +P(3) + P(3) + P(3) \\ \vdots $ The rowwise ...
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Expectation related to geometric random variable

The equation holds for nonnegative random variable. \begin{align} E[Z]&=\sum_{k=1}^\infty kP[Z=k]\\ &=P(Z=1)\\ &+P(Z=2) + P(Z=2)\\ &+P(Z=3) + P(Z=3) + P(Z=3)+\ldots\\ \vdots\\ &=P(...
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Find the distribution of $\frac{X}{\sqrt{X^2+Y^2}}$ where $X$ and $Y$ independent $N(0,1)$

Hint $$Z=\frac{X}{\sqrt{X^2+Y^2}}= \frac{1}{\sqrt{1+W^2}}$$ where $W=Y/X$ are idd standard gaussian. But then $W$ is Cauchy.
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Is the following inequality necessarily true for positive random variables?

As you point in your update, $E(X^2)\le2E^2(X)$ does not holds, in general. In fact: $$E(X^2)\le2E^2(X) \iff E(X^2) - E^2(X) \le E^2(X) \iff Var(X) \le E^2(X) \iff \sigma \le |\mu|$$ Where $\sigma$ is ...
1 vote
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Mean of random variable given a "big" variance

As noted in the comments, for a distribution of variance $\epsilon$ supported on $\{0,1\}$, the mean $p$ solves $$p-p^2=\epsilon \,. \tag{*}$$ Let $$p_1= \frac{1-\sqrt{1-4\epsilon}}2 \in [0, 1/2]$$ ...
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