moment_generating_function

introduction

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MG := Given a random_variable X, it’s K-th moment, 𝔼[X^k], if the expectation is well defined. The MGF is defined as 𝔼[e^tx] ## intuition Used to derive moments and it is an alternative way to find values of a distribution. The moment_generating_function comes from the taylor expansion, so when derivates are taken

Uniqueness. If two rvs have the same moment_generating_function, then they have the same distribution

Properties: - a,b R, Xrv, y= a+bx, ϕ_y(t) = e^atϕ_x(bt) - X,Y independant, $\phi_{X+Y}(t `:=` \mathbb{E}\left[ e^{t(X+Y} \right] = \mathbb{E}\left[ e^{tx}e^{ty} \right] = \phi_X(t)\phi_Y(t)$ which is true for multiple i.d.s

Binomial distribution is just a series of n Bernoulli’s distribution which is expected dude to the ids Properties of a mgf.

Exponential family of equations. - Bernoulli’s, it is a 1-paremeter distribution - Normal - The reason why we go from standard to any normal by $\frac{rv - mean}{standard}$ is because, of the mgf described above, and how the mgf is :todo: ## rigour

exam clinic

1. Let N be a poisson rv with parameter λ > 0, that is $$\mathbb{P}(N=k) = e^{-\lambda} \frac{\lambda^{k}}{k!}$$ find the moment_generating_function Since discrete we can use an easier formula. $\psi_N(t) = \sum _{k=0}^{\infty} e^{-\lambda} \frac{\lambda^{k}}{k!} e^{tk}$ Factor out e^−λ. and the series converges.  = e^λe^tk
2. n independent and identically distributed normal distributions with mean 0 and variance σ². What are the mean vector and the covariance ## examples and non-examples ## resources tags :math:probability_and_statistics_2:

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