MGF Technique - lec. 25

ucla | MATH 170E | 2023-03-19T01:59

Definitions

Let $X_{1}, \dots, X_{n}$ be a seq. of independent r.v.s and $a_{1}, \dots \in \R$

$Y=\sum_ja_jX_j\quad (\text{lin. comb.})\text{has}$

For r.v.s. with MGFs sps. $h > 0$ .t. $t \in (- h, h)$ we have

$M_{X} (t) = M_{Y} (t) ⟹ X, Y independent$

Same MGF → same distribution

For i.i.d. seq. of r.v.s. with common MGF $M (t)$ :

$M_{S_{n}} (t) = [M (t)]^{n} M_{{\overset{―}{X}}_{n}} (t) = [M (\frac{t}{n})]^{n}$

For a seq. of r.v.s. $(X_{n})_{n}^{\infin}$ and another r.v. $X$

$X_n\to X\quad\text{in distribution as}\quad n\to\infin\text{if}$

For a seq. of r.v.s. $(X_{n})_{n}^{\infin}$ and another r.v. $X$

For r.v.s. with MGFs sps. $h > 0$ .t. $t \in (- h, h)$ we have

$M_{X_n}(t)\to M_{X}(T)\quad\text{as}\quad n\to\infin\text{then}$

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