An experiment consists of 6 independent Bernoulli trials with probability of success q in each trial. The random variable X is the number of successes. Enumerate the probability distribution of X. Solution For the binomial distribution, with n trials and p the probability of success in each trial, P(x) = C(n,x)p^x(1-p)^(n-x) In this problem, we are given that n = 6 and the probability of success = q (this is bad notation, because q is normally the probability of non-success. Then, P(x) = C(6,x)q^x(1-q)^(6-x) P(0) = C(6,0)q^0(1-q)^(6-0) = 1 * 1 * (1-q)^6 = (1-q)6 P(1) = C(6,1)q^1(1-q)^(6-1) = 6 * q * (1-q)^5 = 6q(1-q)5 P(2) = C(6,2)q^2(1-q)^(6-2) = 15 * q^2 * (1-q)^4 = 15q2(1-q)4 P(3) = C(6,3)q^3(1-q)^(6-3) = 20 * q^3 * (1-q)^3 = 20q3(1-q)3 P(4) = C(6,4)q^4(1-q)^(6-4) = 15 * q^4 * (1-q)^2 = 15q4(1-q)2 P(5) = C(6,5)q^5(1-q)^(6-5) = 6 * q^5 * (1-q)^1 = 6q5(1-q) P(6) = C(6,6)q^6(1-q)^(6-6) = 1 * q^6 * (1-q)^6 = q6.

1. An experiment consists of 6 independent Bernoulli trials with probability of success q in each
trial. The random variable X is the number of successes. Enumerate the probability distribution
of X.
Solution
For the binomial distribution, with n trials and p the probability of success in each trial,
P(x) = C(n,x)p^x(1-p)^(n-x)
In this problem, we are given that n = 6 and the probability of success = q (this is bad notation,
because q is normally the probability of non-success.
Then, P(x) = C(6,x)q^x(1-q)^(6-x)
P(0) = C(6,0)q^0(1-q)^(6-0) = 1 * 1 * (1-q)^6 = (1-q)6
P(1) = C(6,1)q^1(1-q)^(6-1) = 6 * q * (1-q)^5 = 6q(1-q)5
P(2) = C(6,2)q^2(1-q)^(6-2) = 15 * q^2 * (1-q)^4 = 15q2(1-q)4
P(3) = C(6,3)q^3(1-q)^(6-3) = 20 * q^3 * (1-q)^3 = 20q3(1-q)3
P(4) = C(6,4)q^4(1-q)^(6-4) = 15 * q^4 * (1-q)^2 = 15q4(1-q)2
P(5) = C(6,5)q^5(1-q)^(6-5) = 6 * q^5 * (1-q)^1 = 6q5(1-q)
P(6) = C(6,6)q^6(1-q)^(6-6) = 1 * q^6 * (1-q)^6 = q6