Answer to Question #294517 in Statistics and Probability for Vishnu Kumar

Question

a. Find the pdf of x = [a b] a 2-d vector, where a is a Bernoulli random variable and

b is a Gaussian random variable. Assume, θ is the parameter for Bernoulli which gives

the probability of a = 1, that is p(a=1) = θ. Assume, b follows a Gaussian distribution with

mean m and variance σ

2

. The covariance of x is [1]

[ θ(1-θ) 0;

0 σ

2

]

b. Let the pdf of part (a) be p(x). Now assume that there are N iid samples drawn from

this pdf. Find θ that maximizes the joint probability q(x) of these N samples. Once you

determine q(x), use ln q(x) for computing θ.