. Let X be a random variable. (i) Give an example of a...

. Let $X$ be a random variable. (i) Give an example of a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a random variable $X$ defined on this probability space, and a function $f$ so that the $\sigma$-algebra generated by $f(X)$ is not the trivial $\sigma$-algebra $\{\emptyset, \Omega\}$ but is strictly smaller than the $\sigma$-algebra generated by $X$. (ii) Can the $\sigma$-algebra generated by $f(X)$ ever be strictly larger than the $\sigma$-algebra generated by $X$ ?

. Let $X$ be a random variable.
(i) Give an example of a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a random variable $X$ defined on this probability space, and a function $f$ so that the $\sigma$-algebra generated by $f(X)$ is not the trivial $\sigma$-algebra $\{\emptyset, \Omega\}$ but is strictly smaller than the $\sigma$-algebra generated by $X$.
(ii) Can the $\sigma$-algebra generated by $f(X)$ ever be strictly larger than the $\sigma$-algebra generated by $X$ ?

Key Concepts

Probability Space

A probability space is a triple (?, F, P) consisting of a sample space ?, a sigma?algebra F of subsets of ?, and a probability measure P that assigns a probability to each event in F. This structure serves as the foundational framework for defining and analyzing random phenomena and ensures that all events of interest are measurable and have an associated probability.

Random Variable

A random variable is a measurable function from a probability space to a measurable space (usually the real numbers with the Borel sigma?algebra). This mapping allows us to associate real outcomes to events in the sample space, and its measurability guarantees that preimages of Borel sets are events in F, thereby making probability statements about the outcomes.

Sigma-Algebra

A sigma-algebra is a collection of subsets of a given set (typically the sample space) that is closed under complementation, countable unions, and countable intersections. It provides the mathematical structure necessary to define measure theory, ensuring that operations on events yield sets that are still measurable.

Sigma-Algebra Generated by a Random Variable

The sigma-algebra generated by a random variable X, denoted ?(X), is the smallest sigma-algebra on the sample space that makes X measurable. It is constructed from all preimages of Borel sets under X and encapsulates all the information about the outcomes captured by X.

Measurable Functions and Function Composition

A measurable function (such as f in f(X)) is one for which the preimage of any Borel set is in the underlying sigma-algebra. When composing a random variable with a measurable function (i.e., forming f(X)), the resulting random variable is also measurable. Importantly, the sigma-algebra generated by f(X) is always a subset of ?(X), reflecting potential loss of information if f is not injective.

Injectivity and Information Loss

The concept of injectivity is central when considering the sigma-algebra generated by f(X). If f is injective, then f(X) preserves all the information encoded in X, leading to ?(f(X)) equaling ?(X). However, if f is not injective, some distinctions among the values of X may be 'collapsed', resulting in ?(f(X)) being strictly smaller than ?(X). This idea is key to constructing examples where f(X) indirectly loses some information compared to X, yet retains more than the trivial sigma-algebra.

Trivial Sigma-Algebra

A trivial sigma-algebra on a set consists only of the empty set and the entire set itself. When a measurable function collapses all variability to a single constant, the sigma-algebra generated is trivial. However, in some examples, f is chosen not to be constant to avoid triviality while still being non-injective, ensuring that ?(f(X)) is non-trivial yet strictly smaller than ?(X).

Transcript

00:01 Welcome to numerate.

00:02 In the given problem, we have to argue.

00:05 We have to argue that sigma must be greater than zero.

00:10 First thing is, sigma, as we have known, sigma square is the variance and sigma is the standard deviation.

00:17 It cannot be negative.

00:19 But that we know from our statistical point of view.

00:22 Our statistical point of view always knows that since it's a measure of distance, it has to be greater than zero.

00:30 But how? how? how? will this cumbersome mathematical form will support that.

00:37 So first imagine we know that fy, d, y over the entire range will always give us one.

00:48 Now see the fun here.

00:50 So it is minus infinity to plus infinity, the entire real line.

00:54 1 by sigma root 2 pi e to the power minus half y minus mu by sigma whole square d will be 1 therefore you can think since this is the constant and it is independent of y it can come outside and in the denominator it will remain which can go to the other side being positive.

01:26 So it is minus infinity to plus infinity, e to the power minus half, y minus mu by sigma whole square, d y is equals to sigma root two pi.

01:46 Now, here, we have known that the normal curve, curve is symmetric, right? so, even this function, a bigger one, this also will be symmetric.

02:08 So i can write odd function, even function, so two times 0 to infinity, e to the power minus half y minus mu by sigma whole square...

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