SOA Exam P Questions and Answers 100% Pass

SOA Exam P Questions and Answers 100% Pass Bernoulli Trial - An experiment in which there are exactly two possible outcomes Binomial distribution - A random variable X represents the number of successes observed from the n Bernoulli trials Binomial Parameters - n = number of trials p = probability of success q = 1-p Binomial probability function (f(x)) - E[X] binomial function - E[x] = np var(x) binomial function - var(x) = npq MGF binomial function - 2 Katelyn Whitman, All Rights Reserved © 2025 Additive property of binomial function - Sum of independent binomially distributed variables each with probability p, has parameters of p and the sum of all n Negative binomial distribution - X is the number of failures before r successes in a series of independent Bernoulli trials Negative binomial parameters - r = desired number of successes x = number of failures before r successes p = probability of success q = 1-p, probability of failure Negative binomial probability density function f(x) - f(x) = Pr(X=x) = (r + x - 1)!/x!(r-1)! * p^r * q^x Expected value of negative binomial distribution E[X] - E[X] = rq/p Variance of negative binomial distribution var(x) - var(x) = rq/p^2 Moment generating function negative binomial distribution - Mx(t) = ((1 - qe^t)/p)^-r 3 Katelyn Whitman, All Rights Reserved © 2025 Additive property of negative binomial distribution - If Xi follows a negative binomial distribution with parameters ri and p, and they are independent, then the sum of them follows a negative binomial distribution with parameters as the sum of the ri and p. Geometric distribution - The number of failures observed from the series of Bernoulli trials until the first success occurs