Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

1.2.2 Discrete and Continuous Random Variables

1. 1. Bernoulli Distribution. Let be a Bernoulli random variable, , with probability mass function
   

   Show that and .

   Solution

   If then we can write

   

   and by definition

   

   and hence

   
2. 2. Binomial Distribution. Let be a sequence of independent Bernoulli random variables each with probability mass function

   and let

   Show that follows a binomial distribution, ...