Actuary Exam P Questions and Answers

Bernoulli Trial - ANS--> An experiment in which there are exactly two possible outcomes Binomial distribution - ANS--> A random variable X represents the number of successes observed from the n Bernoulli trials Binomial Parameters - ANS--> n = number of trials p = probability of success q = 1-p Binomial probability function (f(x)) - ANS--> E[X] binomial function - ANS--> E[x] = np var(x) binomial function - ANS--> var(x) = npq MGF binomial function - ANS--> Additive property of binomial function - ANS--> Sum of independent binomially distributed variables each with probability p, has parameters of p and the sum of all n Negative binomial distribution - ANS--> X is the number of failures before r successes in a series of independent Bernoulli trials Negative binomial parameters - ANS--> 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) - ANS--> f(x) = Pr(X=x) = (r + x - 1)!/x!(r-1)! * p^r * q^x Expected value of negative binomial distribution E[X] - ANS--> E[X] = rq/p Variance of negative binomial distribution var(x) - ANS--> var(x) = rq/p^2 Moment generating function negative binomial distribution - ANS--> Mx(t) = ((1 - qe^t)/p)^-r Additive property of negative binomial distribution - ANS--> 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 - ANS--> The number of failures observed from the series of Bernoulli trials until the first success occurs Parameters of geometric distribution - ANS--> p = probability of success q = 1-p x = number of trials before first success Probability density function of geometric distribution - ANS--> f(x) = Pr(X=x) = q^x * p Probability mass function of geometric distribution - ANS--> F(x) = Pr(X <= x) = 1 - q^x+1 Expected value of geometric distribution - ANS--> E[x] = q/p Variance of geometric distribution - ANS--> var(x) = q/p^2 MGF of geometric distribution - ANS--> Mx(t) = ((1 - qe^t)/p)^-1 Additive property of geometric distribution - ANS--> A sum of n independent geometric distributions with parameter p follows a negative binomial distribution with parameters r = n and p. Hypergeometric distribution definition - ANS--> X is the number of Type 1 objects in a sample of n objects randomly selected without replacement from a population of m objects, of which m1 are type 1 and m2 are type 2 (where m = m1 + m2. Hypergeometric probability distribution function - ANS--> f(x) = Pr(X=x) = C(m1,x) * C(m2,n-x) = C(m,n) Expected value of hypergeometric function - ANS--> E[x] = nm1/m Variance of hypergeometric function - ANS--> var(x) = n(m1/m)(m2/m)((m-n)/(m-1)) Definition of poisson distribution - ANS--> X is the number of occurrences of some "rare" event in a unit time period where λ is the rate of occurrences per unit time period Probability distribution function - ANS--> f(x) = Pr(X=x) = e^-λ*λ^x/x! Expected value of a poisson distribution - ANS--> λ Variance of poisson distribution - ANS--> λ MGF of poisson distribution - ANS--> Mx(t) = e^(λ((e^t)-1)) Additive property of poisson distribution - ANS--> Sum of poisson distributions has parameter of the sum of all the λ Uniform distribution - ANS--> Constant probability density function on the interval [a,b] PDF of uniform distribution - ANS--> f(x) = 1/(b-a) CDF of uniform distribution - ANS--> F(x) = (x-a)/(b-a) Expected value of uniform distribution - ANS--> E(x) = (b+a)/2 Variance of uniform distribution - ANS--> var(x) = ((b-a)^2)/12 median of uniform distribution - ANS--> m = (b+a)/2 MGF of uniform distribution - ANS--> Mx(t) = (e^bt - e^at)/t(b-a) exponential distribution parameters - ANS--> θ Exponential distribution CDF - ANS--> F(x) = 1-e^(-x/θ) Expected value of exponential distribution - ANS--> θ variance of exponential distribution - ANS--> θ^2 median of exponential distribution - ANS--> m = θ*ln(2) moment generating function for exponential distribution - ANS--> M(t) = 1/(1-θt) for t < 1/θ memoryless property for exponential distribution - ANS--> Pr(X > x0 + x | x > x0) = Pr(X>x) Exponential distribution PDF - ANS--> f(x) = (1/θ) e^(-x/θ) Gamma distribution - ANS--> Gamma distribution function - ANS--> expected value of gamma distribution - ANS--> E[x] = αθ variance of gamma distribution - ANS--> var(x) = αθ^2 moment generating function of gamma distribution - ANS--> M(t) = 1/((1- θt)^α) Additive property of gamma distribution - ANS--> Sum of gamma distributions has parameters of the sum of the α and θ. Normal distribution expected value - ANS--> μ Normal distribution variance - ANS--> σ^2 The central limit theorem - ANS--> If X1...Xn are independent and identically distributed with mean μ and variance σ^2. For large n, then the sum of X1...Xn is approximately distributed with mean nμ and variance nσ^2. The mean of all of them is distributed with mean μ and variance σ^2/n Joint probability function - ANS--> Pr(X=x and Y=y) = fxy(x,y) Conditional probability function of X fx(x)(multivariate) - ANS--> fx(x|Y=y) = fxy(x,y)/fy(y) Marginal probability density function of X - ANS--> E[g(x,y)] - ANS--> sum or integral of g(x,y)fxy(x,y) Mx,y(s,t) - ANS--> E[e^sX+tY] cov(X,Y) - ANS--> E[XY] - E[X]E[Y] Covariance function properties - ANS--> var(X) = cov(X,X) cov(X,Y) = cov(Y,X) cov(aX + bY,z) = a cov(X,Z) + b cov(Y,Z) cov(X,Y) = 0 if X and Y are independent var(aX + bY) = cov(aX+bY,aX+bY) = a^2 var(X) + b^2 var(Y) + 2abcov(X,Y) Correlation coefficient - ANS--> rho cov(X,Y)/sd(X)sd(Y) bivariate normal distribution conditional distribution of x - ANS--> E[X|Y=y] = μx + ρ(σy/σx)(y-μy) var(X|Y=y) = σx^2(1-ρ^2)