# Tag Info

### How to find the distribution of a function of two random variables?

Often, the most straightforward way to find the distribution of a variable defined in terms of other random variables is to compute its cumulative distribution function. For any number $y$ this ...
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1 vote

### On Definition of Random Variables

To quote from Wikipedia, a $\sigma$-algebra is defined as $\Omega$ is in $\mathcal F$, and $\Omega$ is considered to be the universal set in the following context. $\mathcal F$ is closed under ...
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1 vote
Accepted

As @whoknowsnot has shown, for every $a \in \mathbb R$, the event $\{X< a\}\in \mathcal F$, and since $\{X\leq a\}\in \mathcal F$ by definition, the difference of these two events, namely \{X = ... 7 votes ### Expectation of the product of iid random variables There are several things to note here. Multiplying lots of numbers in computer leads to rounding errors, especially if these numbers vary in magnitude, i.e. small numbers are multiplied by large ... • 33.5k 10 votes ### Expectation of the product of iid random variables As an addendum to @whuber's explanation, consider that \begin{align} \prod_{i=1}^N \vert X_i\vert &= \exp\{ \sum\nolimits_{i=1}^N \overbrace{\ln \vert X_i\vert}^{Y_i} \}\\ &= \exp\{ \sum\... • 92.7k 24 votes Accepted ### Expectation of the product of iid random variables First, let's establish the correct identity. WhenX_1, \ldots, X_N$are independent variables with finite expectations$\mu_i=X_i,\$ then by laws of conditional expectation, $$E\left[\prod_{i=1}^N X_i\... • 291k 0 votes ### What is the "opposite" of a random variable? I would say a deterministic variable. Examples: Random variable- the amount of heads when a coin is tossed 100 times. Deterministic variable- the age of the Eifel tower in exactly 12 years from now. 0 votes ### Can two random variables be independent in some basis and dependant in other? Only a hint: Your question is ambiguous, it admits of different interpretations, as verified by the comments of users Dilip Sarwate and whuber. For the specific question about transformation to polar ... • 65.8k 1 vote ### Can two random variables, both of which are dependent on a 3rd random variable, be independent of each other? This is possible. Two examples: Discrete example Throw two coins independently, but code the outcomes as 0 or 1. Then define:$$ \begin{align} X&=\text{outcome on first coin}\\ Y&...
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