56 Basic Concepts of Mixture Models

Exercise 56.1

Which one of the following is not the density of a well defined mixture distribution with support on x \ge 1 x

f(x) = \frac{1}{2}\ e^{-x} + \frac{1}{2} \frac{1}{\sqrt{2 \pi}} \exp^{-0.5x^2}
f(x) = \frac{1}{2}\ e^{-x} + \frac{1}{4}\ e^{-x}
f(x) = \frac{1}{2}\ e^{-x} + \frac{1}{2}\ e^{- 0.5 x}

Hint: the key here is to write the mixtures with the weights and the normalization constant clearly separated. This reveals that the last one is not a well defined mixture distribution because the weights do not sum to 1 while the the second answer is!

Exercise 56.2

What is the expected value of a random variable X whose distribution is a mixture of Poisson distributions of the form

f(x) = 0.3 \frac{2^x e^{-2}}{x!} + 0.45 \frac{2^x e^{-3}}{x!} + 0.25 \frac{.5^x e^{-0.5}}{x!} \tag{56.1}

Solution:

E(X) = 0.3 \cdot 2 + 0.45 \cdot 3 + 0.25 \cdot 0.5 = 2.075

Exercise 56.3

What is the variance of an RV X whose distribution is a mixture of Poisson distributions of the form Equation 56.1 ?

E(X^2) = 0.3 \cdot (2+2^2 ) + 0.45 \cdot (3+3^2) + 0.25 \cdot (0.5 + 0.5^2)= 7.3875

Solution:

Var(X) = E(X^2) - E(X)^2 = 7.3875 - (2.075)^2 = 3.081875