# Tag Info

### 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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### 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 ...
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### 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 ...
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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 ...
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### 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
Accepted

### 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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