Lecture - Probability MT22, X

Created: November 11, 2022

Flashcards

If $X _ i \sim \text{Poi}(\lambda _ i)$, and all $X _ i$ are mutually independent, what $G _ {\sum^n _ {i=1} X _ i}(s)$?
\[\prod_{i=1}^n G_{X_i} (s)\]

If $X _ 1, X _ 2, X _ 3$ are identically distributed and independent, are $X _ 1 + X _ 2 + X _ 3$ and $3X _ 1$ the same?

No.

What is the $m$-th moment of a random variable $X$?
\[\mathbb{E}[X^m]\]

If $X \sim \text{Poi}(\lambda)$, what is $G _ X(s)$?
\[e^{\lambda (s - 1)}\]

If $X \sim \text{Ber}(p)$, what is $G _ X(s)$?
\[1 - p + ps\]

If $X \sim \text{Geom}(p)$, what is $G _ X(s)$?
\[\frac{ps}{1 - (1-p)s}\]

If $X \sim \text{Bin}(n, p)$, what is $G _ X(s)$?
\[((ps) + (1-p))^n\]

What are the conditions on the $X _ i$s in the random sums theorem about $R = \sum^N _ {i=1} X _ i$?

They are identically distributed.

What are the conditions on the $X _ i$s and $N$ in the random sums theorem about $R = \sum^N _ {i=1} X _ i$?

They are independent.

Let $X _ i, i \ge 1$ be identically distributed random variables with p.g.f. $G _ X(s)$ and let $N$ be another random variable indepedent of all $X _ i$. What is the p.g.f. of $R = \sum^N _ {i=1} X _ i$?
\[G_R(s) = G_N(G_X(s))\]


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