SOA EXAM P QUESTIONS AND
VERIFIED ANSWERS|100%
CORRECT|GRADE A+
Bernoulli Trial - ANSWER An experiment in which there are exactly two possible outcomes
Binomial distribution - ANSWER A random variable X represents the number of successes observed from
the n Bernoulli trials
Binomial Parameters - ANSWER n = number of trials
p = probability of success
q = 1-p
Binomial probability function (f(x)) - ANSWER
E[X] binomial function - ANSWER E[x] = np
var(x) binomial function - ANSWER var(x) = npq
MGF binomial function - ANSWER
Additive property of binomial function - ANSWER Sum of independent binomially distributed variables
each with probability p, has parameters of p and the sum of all n
Negative binomial distribution - ANSWER X is the number of failures before r successes in a series of
independent Bernoulli trials
Negative binomial parameters - ANSWER 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) - ANSWER f(x) = Pr(X=x) = (r + x - 1)!/x!(r-1)! * p^r *
q^x
Expected value of negative binomial distribution E[X] - ANSWER E[X] = rq/p
Variance of negative binomial distribution var(x) - ANSWER var(x) = rq/p^2
Moment generating function negative binomial distribution - ANSWER Mx(t) = ((1 - qe^t)/p)^-r
Additive property of negative binomial distribution - ANSWER 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 - ANSWER The number of failures observed from the series of Bernoulli trials
until the first success occurs
Parameters of geometric distribution - ANSWER p = probability of success
q = 1-p
x = number of trials before first success
Probability density function of geometric distribution - ANSWER f(x) = Pr(X=x) = q^x * p
Probability mass function of geometric distribution - ANSWER F(x) = Pr(X <= x) = 1 - q^x+1
Expected value of geometric distribution - ANSWER E[x] = q/p
Variance of geometric distribution - ANSWER var(x) = q/p^2
VERIFIED ANSWERS|100%
CORRECT|GRADE A+
Bernoulli Trial - ANSWER An experiment in which there are exactly two possible outcomes
Binomial distribution - ANSWER A random variable X represents the number of successes observed from
the n Bernoulli trials
Binomial Parameters - ANSWER n = number of trials
p = probability of success
q = 1-p
Binomial probability function (f(x)) - ANSWER
E[X] binomial function - ANSWER E[x] = np
var(x) binomial function - ANSWER var(x) = npq
MGF binomial function - ANSWER
Additive property of binomial function - ANSWER Sum of independent binomially distributed variables
each with probability p, has parameters of p and the sum of all n
Negative binomial distribution - ANSWER X is the number of failures before r successes in a series of
independent Bernoulli trials
Negative binomial parameters - ANSWER 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) - ANSWER f(x) = Pr(X=x) = (r + x - 1)!/x!(r-1)! * p^r *
q^x
Expected value of negative binomial distribution E[X] - ANSWER E[X] = rq/p
Variance of negative binomial distribution var(x) - ANSWER var(x) = rq/p^2
Moment generating function negative binomial distribution - ANSWER Mx(t) = ((1 - qe^t)/p)^-r
Additive property of negative binomial distribution - ANSWER 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 - ANSWER The number of failures observed from the series of Bernoulli trials
until the first success occurs
Parameters of geometric distribution - ANSWER p = probability of success
q = 1-p
x = number of trials before first success
Probability density function of geometric distribution - ANSWER f(x) = Pr(X=x) = q^x * p
Probability mass function of geometric distribution - ANSWER F(x) = Pr(X <= x) = 1 - q^x+1
Expected value of geometric distribution - ANSWER E[x] = q/p
Variance of geometric distribution - ANSWER var(x) = q/p^2
