Lecture - Probability MT22, X

Flashcards

If $X_{i} \sim Poi (λ_{i})$ , and all $X_{i}$ are mutually independent, what $G_{\sum_{i = 1}^{n} X_{i}} (s)$ ?

\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 $3 X_{1}$ the same?

No.

What is the $m$ -th moment of a random variable $X$ ?

E [X^{m}]

If $X \sim Poi (λ)$ , what is $G_{X} (s)$ ?

e^{λ (s - 1)}

If $X \sim Ber (p)$ , what is $G_{X} (s)$ ?

1 - p + p s

If $X \sim Geom (p)$ , what is $G_{X} (s)$ ?

\frac{p s}{1 - (1 - p) s}

If $X \sim Bin (n, p)$ , what is $G_{X} (s)$ ?

((p s) + (1 - p))^{n}

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

They are identically distributed.

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

They are independent.

Let $X_{i}, i \geq 1$ 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 $R = \sum_{i = 1}^{N} X_{i}$ ?

G_{R} (s) = G_{N} (G_{X} (s))

Flashcards

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