2022-10-06
The Moment Generating Functions for DRVs is defined as
\[M_X(s) = E\left[e^{sX}\right] = \sum_{k=0}^{\infty} E\left[X^k \right] \frac{s^k}{k!}.\]
We can obtain all moments of \(X_k\) from its MGF: \[M_X(s) = \sum_{k=0}^{\infty} E\left[X^k\right] \frac{s^k}{k!},\]
and we can write the \(k\)th moment as the \(k\)th derivative of the MGF evaluated at \(s = 0\). \[E\left[X^k\right]= \frac{dk}{ds^k} M_X(s) \Big|_{s=0}.\]
Moment generating functions are important for a variety of reasons, one of which is that they may be used to analyze sums of random variables.
The kth moment of a random variable \(X\) is defined to be \(E\left[X^k\right]\).
The kth central moment of \(X\) is defined to be \(E\left[\left(X−E[X]\right)^k\right]\).
The variance of a random variable measures the spread of the variable around its expected value.
[PSDR] Definition 3.31 Let X be a random variable with expected value \(\mu = E[X]\). The variance of \(X\) is defined as
\[Var(X) = E\left[(X−\mu)^2\right]\]
The standard deviation of \(X\) is written \(\sigma(X)\) and is the square root of the variance:
\[\sigma(X) = \sqrt{Var(X)}.\]
[PSDR] Theorem 3.9 \[\begin{align*} Var(X) = & E\left[(X - E[X])^2\right] \\ = & E\left[X^2\right] − E[X]^2 \end{align*}\]
Let \(X \sim Binom(3,0.5)\). Compute the standard deviation and variance of \(X\).
The first moment is \(E[X] = np\) (mean) for a Binomial random variable. We need to find the second moment \(E[X^2]\).
Suppose that the MGF is \(M_X(s) = (1- p + pe^s)^n\). Then,
\[E[X^2] = \frac{d2}{ds^2} M_X(s) \Big|_{s=0} = np(np-p+1) = n^2p^2 - np^2 + np.\]
So,
\[\begin{align*} Var(X) = & E\left[X^2\right] − E[X]^2 \\ = & n^2p^2 - np^2 + np - n^2p^2 \\ = np(1-p) \\ \end{align*}\]
\[\sigma(X) = \sqrt{np(1-p)}\]
\[\sigma = \sqrt{3(0.50)(0.50)}\]
[PSDR] Theorem 3.10
\[Var(cX) = c^2 Var(X)\] \[\sigma(cX) = |c|\sigma(X)\]
\[Var(X_1 + X_2 + \cdots + X_n) = Var(X_1) + Var(X_2) + \dots + Var(X_n)\]