Question

A coin that lands on heads with a probability of p is tossed multiple times. Each toss is independent. X is the number of heads in the first m tosses and Y is the number of heads in the first n tosses. m and n are fixed integers where 0 < m < n. Find the joint distribution of X and Y.

Answer 1

Given: A coin is tossed with probability p.
Each toss is independent
X: the number of heads in first m tosses
  Y: the number of heads in first n tosses
0<m<n

Answer:    

The experiment of getting a head or a tail on tossing a coin is a bernoulli trial.

The probability of getting a head is p.
the probability of getting a tail is 1-p.

Say for random variable X, the experiment is repeated m times. Thus by definition, we have X follows binomial distribution with parameter m and p

Similarly for random variable Y, the experiment is repeated n times, thus Y follows binomial distribution with parameters n and p.

Now, we have the tosses are independent thus X and Y is also independent.

The joint distribution f(x,y) is given by
f(x,y)= f(x) . f(y) as both X and Y are independent

that is the joint distribution is product of distributions of random variable X and random variable Y which are both binomial distribution.