Lecture - Probability MT22, XVI

Created: December 14, 2022

Flashcards

What is

\[\text{Var}\left( \sum^n _ {i=1} X _ i \right)\]

?
\[\sum _ {i=1}^n \text{Var}(X _ i) + 2\sum _ {1 \le i < j \le n} \text{Cov}(X _ i, X _ j)\]

What is a random sample of a distribution?

Independent random variables $X _ 1, X _ 2, \ldots, X _ n$ with the same distribution.

Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$, what is the sample mean $\overline{X _ n}$?
\[\frac{1}{n} \sum^n_{i=1} X_i\]

Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$, what is the sample mean $\mathbb{E}[\overline{X _ n}]$?
\[\mu\]

Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$, what is the sample mean $\text{Var}(\overline{X _ n})$?
\[\frac{\sigma^2}{n}\]

Let $X _ 1, \ldots, X _ n$ be a random sample (i.i.d. r.v.s.) with mean $\mu$. What does the weak law of large numbers state for any $\epsilon > 0$ (as $n \to \infty$)?
\[\lim_{n \to \infty}\mathbb{P}\left(\left|\frac{1}{n} \sum^n_{i=1} X_i - \mu\right| > \epsilon\right) = 0\]

For an event $A$, what is the definition of the random variable $\mathbb{1} _ A$?
\[\mathbb{1}_A = \begin{cases} 1 \text{ if } A_i \text{ occurs} \\\\0 \text{ otherwise} \end{cases}\]

What does Markov’s inequality state about a non-negative r.v. $Y$ with finite expectation?
\[\forall t> 0, \text{ } \mathbb{P}(Y \ge t) \le \frac{\mathbb{E}[Y]}{t}\]

What random variable do you consider in the proof of Markov’s inequality?
\[\tilde{Y_t} = \begin{cases}0 \text{ if } Y < t \\\\t \text{ if } Y \ge t\end{cases}\]

What does Chebyshev’s inequality state about a r.v. $Z$ with finite variance?
\[\forall c>0, \text{ } \mathbb{P}(|Z - \mathbb{E}[Z]| \ge c) \le \frac{\text{Var}(Z)}{c^2}\]

The weak law of large numbers states that

\[\lim _ {n \to \infty}\mathbb{P}\left(\left \vert \frac{1}{n} \sum^n _ {i=1} X _ i - \mu\right \vert > \epsilon\right) = 0\]

What is the interpretation of what this means?
\[\overline{X_n} \approx \mu \text{ for large }n.\]


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