8. Probability Distributions as Modeling Tools

Question 1

X ~ Bernoulli(p)

Answer 1

• P(success) = p
• Mean = p
• Variance = p * (1 - p)

Question 1

X ~ Geometric(p)

Answer 1

• P(success) = p
• f(x) = (1 - p)^x * (p)
• Mean = (1 - p) / p
• Variance = (1 - p) / (p^2)

Question 2

X ~ Binomial(n, p)

Answer 2

• P(success) = p
• f(x) = (n choose x) * (p)^x * (1 - p)^(n-x)
• Mean = np
• Variance = np * (1 - p)

Question 3

X ~ Poisson(lambda)

Answer 3

• f(x) = e^(-lambda) * lambda^x / x!
• Mean = lambda
• Variance = lambda

Question 4

X ~ Exp(lambda)

Answer 4

• f(x) = lambda * e^(-lambda * x)
• F(x) = 1 - e^(-lambda * x)
• Mean = 1 \ lambda
• Variance = 1 \ lambda^2

Question 5

X ~ Unif(a, b)

Answer 5

• Mean = (a + b) / 2
• Variance = (a - b)^2 / 12