SOA EXAM P QUESTIONS AND
CORRECT ANSWERS
var(x) binomial function ANSW✅✅var(x) = npq
MGF binomial function ANSW✅✅
Additive property of binomial function ANSW✅✅Sum of independent binomially distributed
variables each with probability p, has parameters of p and the sum of all n
Negative binomial distribution ANSW✅✅X is the number of failures before r successes in a series
of independent Bernoulli trials
Negative binomial parameters ANSW✅✅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) ANSW✅✅f(x) = Pr(X=x) = (r + x - 1)!/x!(r-1)! *
p^r * q^x
Expected value of negative binomial distribution E[X] ANSW✅✅E[X] = rq/p
Variance of negative binomial distribution var(x) ANSW✅✅var(x) = rq/p^2
Moment generating function negative binomial distribution ANSW✅✅Mx(t) = ((1 - qe^t)/p)^-r
Bernoulli Trial ANSW✅✅An experiment in which there are exactly two possible outcomes
Binomial distribution ANSW✅✅A random variable X represents the number of successes
observed from the n Bernoulli trials
, Binomial Parameters ANSW✅✅n = number of trials
p = probability of success
q = 1-p
Binomial probability function (f(x)) ANSW✅✅
E[X] binomial function ANSW✅✅E[x] = np
Additive property of negative binomial distribution ANSW✅✅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 ANSW✅✅The number of failures observed from the series of Bernoulli
trials until the first success occurs
Parameters of geometric distribution ANSW✅✅p = probability of success
q = 1-p
x = number of trials before first success
Probability density function of geometric distribution ANSW✅✅f(x) = Pr(X=x) = q^x * p
Probability mass function of geometric distribution ANSW✅✅F(x) = Pr(X <= x) = 1 - q^x+1
Expected value of geometric distribution ANSW✅✅E[x] = q/p
Variance of geometric distribution ANSW✅✅var(x) = q/p^2
MGF of geometric distribution ANSW✅✅Mx(t) = ((1 - qe^t)/p)^-1
Additive property of geometric distribution ANSW✅✅A sum of n independent geometric
distributions with parameter p follows a negative binomial distribution with parameters r = n and p.
CORRECT ANSWERS
var(x) binomial function ANSW✅✅var(x) = npq
MGF binomial function ANSW✅✅
Additive property of binomial function ANSW✅✅Sum of independent binomially distributed
variables each with probability p, has parameters of p and the sum of all n
Negative binomial distribution ANSW✅✅X is the number of failures before r successes in a series
of independent Bernoulli trials
Negative binomial parameters ANSW✅✅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) ANSW✅✅f(x) = Pr(X=x) = (r + x - 1)!/x!(r-1)! *
p^r * q^x
Expected value of negative binomial distribution E[X] ANSW✅✅E[X] = rq/p
Variance of negative binomial distribution var(x) ANSW✅✅var(x) = rq/p^2
Moment generating function negative binomial distribution ANSW✅✅Mx(t) = ((1 - qe^t)/p)^-r
Bernoulli Trial ANSW✅✅An experiment in which there are exactly two possible outcomes
Binomial distribution ANSW✅✅A random variable X represents the number of successes
observed from the n Bernoulli trials
, Binomial Parameters ANSW✅✅n = number of trials
p = probability of success
q = 1-p
Binomial probability function (f(x)) ANSW✅✅
E[X] binomial function ANSW✅✅E[x] = np
Additive property of negative binomial distribution ANSW✅✅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 ANSW✅✅The number of failures observed from the series of Bernoulli
trials until the first success occurs
Parameters of geometric distribution ANSW✅✅p = probability of success
q = 1-p
x = number of trials before first success
Probability density function of geometric distribution ANSW✅✅f(x) = Pr(X=x) = q^x * p
Probability mass function of geometric distribution ANSW✅✅F(x) = Pr(X <= x) = 1 - q^x+1
Expected value of geometric distribution ANSW✅✅E[x] = q/p
Variance of geometric distribution ANSW✅✅var(x) = q/p^2
MGF of geometric distribution ANSW✅✅Mx(t) = ((1 - qe^t)/p)^-1
Additive property of geometric distribution ANSW✅✅A sum of n independent geometric
distributions with parameter p follows a negative binomial distribution with parameters r = n and p.
