Write down (but don’t evaluate) an integral that represents the kurtosis of a normal distribution with mean 70 and standard deviation 3.
Write down (but don’t evaluate) a sum that represents the 3rd moment of a Geometric random variable with parameter \(p\).
Find the moment generating function for \(X \sim \text{Unif}(a,b)\).
The moment generating function of \(Y \sim \text{Binom}(1,p)\) is \(m_Y(t) = (q+pe^t)\) where \(q = 1-p\). Re-write this moment generating function as a Taylor series centered at \(t=0\). Write your answer in both \(\Sigma\)-notation and term-by-term.
Suppose that \(Y\) is a linear transformation of a random variable \(X\) given by \(Y = a + bX\), where \(a, b\) are constants. Show that the moment generating function of \(Y\) is \(m_Y(t) = e^{at} m_X(bt)\).
For \(Z \sim \text{Norm}(0,1)\), we saw that the moment generating function of \(Z\) is \(e^{t^2/2}\). If \(X \sim \text{Norm}(\mu,\sigma)\), use the fact that \(X\) can be expressed as \(X = \mu + \sigma Z\) to find the moment generating function for \(X\).
If \(X \sim \text{Pois}(\lambda)\), then the moment generating function for \(X\) is: \(e^{\lambda(e^t-1)}\). Use this and the fact that probability distributions are completely determined by their moment generating functions to show that the sum of two independent Poisson random variables has a Poisson distribution.
The moment generating function of a \(\text{Binom}(n,p)\) random variable is: \((pe^t + 1 - p)^n\). Differentiate this function twice to find the 1st and 2nd moments of a binomial random variable in terms of \(n\) and \(p\).
Use MGFs to prove that the sum of any two independent normal random variables has a normal distribution, even if the two random variables don’t have the same mean or variance.
The triangle distribution is the continuous probability distribution on the interval \([0,2]\) with the density function \[f(x) = \begin{cases} x & \text{for } 0 \le x \le 1, \\ 2-x & \text{for } 1 < x \le 2. \end{cases}\] Use the definition of a MGF to find the MGF for the triangle distribution. It’s okay to use a computer to calculate the integral.
Use MGFs to prove that sum of two independent \(\text{Unif}(0,1)\) random variables has the triangle distribution.