Exam P for actuaries

Explains the concept of a random variable, its pdf and cdf, and distinguishes discrete from continuous variables with examples like coin toss and die roll.

Explore discrete random variables, and learn to read their pdf and cdf through examples of a fair die and a uniform 1 to 100 variable.

Derive the pdf for y, the number of heads in three coin tosses, with y in {0,1,2,3} and probabilities 1/8, 3/8, 3/8, 1/8, then outline the cdf.

Explore the discrete random variable A, the number of coin tosses until the first head, with pdf P(A=a)=(1/2)^a and its corresponding cdf.

Apply discrete probability: verify pdf properties, analyze a loaded die with probability proportional to j, and compute the even-face probability; use a geometric model to find P(X is even) 1/3.

The lecture computes probabilities for two continuous variables: with f_x = 2x on [0,1], P(0.5 < x < 1) = 3/4 and P(0.5 < x < 0.75) = 5/16, and with f_x = 3 - 48x^2 on [-0.25, 0.25], P(1/8 < x < 5/16) = 5/32.

Explore properties of continuous random variables, showing how the area under the pdf gives P(a ≤ X ≤ b) and P(X = a) = 0, while f(a) can exceed 1.

Explore the CDF for continuous and discrete cases, where f(t) equals the left-side probability, and the survival function s(t) equals the right-side probability x > t, without an equal sign.

Learn how the cdf and survival function behave as x tends to infinity or minus infinity, with sample pdfs 2x on [0,1] and e^{-y} for y>0.

Hazard rate, also called hazard function or source of mortality, is defined as h(a) = f(a) / S(a), linking time of death density to survival probability via Bayes theorem.

Explore hazard rate in a discrete death-time model with ten end-of-year spots, derive f(a), s(a), and h(a) using a 0.1 probability per spot, and compare formulas and reasoning.

Explore hazard rate with a continuous uniform model on [0,10], computing P(X=1|X≥1)=1/9 and P(X=5|X≥5)=1/5, and derive h(a)=1/(10−a) via Bayes and reasoning.

Compute the hazard rate h(x) = f(x)/S(x) using the survival function S(x) = ∫_x^∞ 0.01 e^{-0.01 t} dt = e^{-0.01 x}. This yields h(x) = 0.01.

Examine the relations among pdf, cdf, and survival function: the pdf integrates to one, the cdf is its integral, and hazard rate equals pdf over the survival function.

Differentiate the cdf 1/(1+e^{-x}) to obtain the pdf f(x)= e^{-x}/(1+e^{-x})^2, and analyze a mixed distribution with a jump at 1 (mass) and a tail for t>1.

this quiz analyzes whether A (x is even) and B (x multiple of three) are independent, using pdf and geometric series to compare P(A∩B) with P(A)P(B).

In exam p for actuaries, quiz 4 derives the pdfs of the max and min of four iid variables with pdf 2x on [0,1], yielding f_y(y)=8y^7 and f_z(z)=8z(1−z^2)^3.

Compute the probability that at least two import cars pass before third domestic car using analysis: a = one import, b = zero imports; p(import) = 0.25, p(domestic) = 0.75.

Explore expected value, or mean, of a random variable in discrete and continuous cases. Learn to compute e of x with sums for discrete distributions and integrals of x f(x).

Compute the expectation for discrete and continuous random variables using sums and integrals. See why E[h(X)] differs from h(E[X]) with examples like dice payoffs and a continuous pdf.

Learn the terminology of raw moments and central moments, including mu as the mean and the second central moment for a die, the expectation yielding 35/12.

The lecture analyzes the number of fair-die tosses needed to obtain the first 1, derives P(X=a) = (5/6)^{a-1}(1/6) for a ≥ 1, and shows the expected value is six.

Compute the expected value of 0.5^x for the first occurrence of a one when tossing a fair die, using the geometric distribution with p=1/6 and r=5/12, yielding 1/7.

Explore the variance of a random variable, its interpretation as dispersion, and derive the formula Var(X)=E[X^2]-(E[X])^2 using the second central moment.

Explore variance and the difference between E[x^2] and (E[x])^2 using two dice, A with faces 1–6 and B with 1,1,1,6,6,6; A’s variance ≈ 2.91 and B’s ≈ 6.25.

Compute the mean and variance of two symmetric triangular densities: x with f(x)=1-|x| on [-1,1] and w with f(w)=0.5-0.25|w| on [-2,2], yielding E[x]=E[w]=0, Var(x)=1/6, Var(w)=2/3.

Explore variance properties, showing that adding a constant leaves variance unchanged and multiplying by a constant scales it by the square, illustrated with uniform x and y.

Compute the moment generating function of an exponential(5) variable by integrating e^{t x} times its pdf over x>0. This yields M_X(t)=5/(5 - t) for t < 5.

Learn the moment generating function basics: m_x(0)=1 and moments come from derivatives at zero; compute variance from the log’s second derivative, and use m_x(t)=5/(5−t) (t<5) to get E[X]=1/5, E[X^2]=2/25, Var(X)=1/25.

Compute the variance of x from its moment generating function m_x(t) = alpha/(alpha - t) by taking the natural log, differentiating twice, and evaluating at t = 0, yielding 1/alpha^2.

Explore the probability generating function for a discrete random variable, denoted p_x(t), using derivatives at zero to extract probabilities and derivatives at one to obtain moments like E[X] and E[X(X-1)].

Explore how percentile defines distribution cutoffs using the CDF and survival function, with examples of the 50th, 80th, and general p percentiles, and a 87.5th percentile calculation.

Compute the 50th and 60th percentile of the density f(x)=2/15(4−x) on [0,3] by integrating to set the CDF to 0.5 and 0.6, then solve the resulting quadratic equations.

Analyze a discrete variable with f_x = e^{-2} 2^x / x! by building the cdf from x = 0,1,2,… to find 50th and 80th percentiles: 2 and 3.

Identify the mode of a distribution by finding stationary points with f'(x)=0 and evaluating endpoints, then compare f(x) values; for the given density on [0,3], the mode is x=2.

Derive the minimum of three independent, identically distributed exponential variables with rate 1/10, yielding S(a)=e^{-3a/10} and f(a)=(3/10)e^{-3a/10} for a>0.

Compute the system lifetime as the minimum of seven iid lifetimes with density f(x)=4/x^3 for x>1. Derive the survival and pdf, then estimate the expected lifetime, about 138 units.

Explore the maximum of independent identically distributed variables and its cdf, F_X(a)^n. Compute the corresponding pdf by differentiating the cdf, and illustrate with a uniform 0–20 example.

Explore the input–output relationships across numbers, functions, integration, expectation, variance, and the moment generating function, and identify what counts as input and what counts as output.

Identify key notes on expectations and variance, including non-existent means, variance under constants and independence, and symmetry about a point (with a brief note on skewness).

Compute the expected value of a 1 million blind trust growing at 10% as the president risks reelection, using an infinite geometric series to model the first loss.

Compute the expected difference between the maximum and minimum of three iid uniform(0,1) variables. Apply the CDF and survival function to derive E[max]=3/4 and E[min]=1/4, yielding E[max-min]=1/2.

From the moment generating function, infer that x takes values 0, 1, and 2, and derive the pdf with probabilities 1/6, 2/6, and 3/6.

Calculate the expected payout by evaluating E(g(X)) with X as the first-head toss. Show E[(1/2)^X] = 1/3, so E(g(X)) = 2000, and conclude that paying 2100 to play is unfavorable.

Explore the discrete uniform distribution with n outcomes from 1 to n, each with probability 1/n, and learn its pdf, cdf, and the formulas for expectation and variance.

Transform a discrete uniform on even numbers from 0 to 22 into a standard uniform by computing x/2+1. Compute its expectation and variance, yielding E[x]=11 and Var[x]=143/3.

Explore the binomial distribution with parameters n and p, modeling the number of successes in n independent trials (X ~ Bin(n,p)). See a three coin toss example to illustrate outcomes.

Explore the binomial distribution with n = 3 and p = 0.7, deriving P(X = 0,1,2,3) and summing scenarios to obtain the pdf for x.

Derive the binomial distribution for n trials with success probability p and failure q = 1 − p, and express the pmf as C(n, x) p^x q^{n − x} for x from 0 to n.

Understand the relationship between Bernoulli and binomial distributions, and memorize the binomial expectation np and variance np(1-p) for exam readiness.

Explore binomial distribution with n=5 and p=0.2 by comparing two random test takers' scores and the probability they match, then the probability their papers are identical.

Explore the binomial distribution with 72 dice throws to find E[X^2] for number of sixes. Use the variance formula npq and expected value equals np to obtain E[X^2] equals 154.

Explore the Poisson distribution, a counting model for events per unit time with parameter lambda. Use its PMF e^-λ λ^x / x!, and note the mean and variance equal lambda.

Apply the Poisson distribution to baseball hits and claims. Derive lambda from a no-hit probability, compute the probability of four or more hits, and find the variance.

Model rainy days with a Poisson(0.6) distribution, determine insurance payouts of 0, 1000, or 2000, and compute the payment's standard deviation.

Explore Poisson distribution properties, showing how lambda scales with time and event probability. Use the formula e^-lambda lambda^x/x! to compute arrivals per hour, per two hours, and per fifteen minutes.

Explore the geometric distribution with parameter p, the probability of success, including two forms: number of failures before first success and number of trials until first success, with pdfs.

Explore the geometric distribution by calculating the probability the first success occurs on trial x for a fair die, and find the smallest a with P(X < a) > 1/2.

Explore negative binomial distribution with parameters r and p, modeling the number of failures until r successes or the number of trials until r successes, with r=1 yielding geometric distribution.

Explore the expectation and variance of the negative binomial variables x (failures) and y (trials). Derive E[X]=rq/p, Var(X)=rq/p^2, E[Y]=r/p, Var(Y)=rq/p^2, and the relation Y=X+r.

apply the negative binomial framework to find the probability that the third 'one' occurs on the nth toss of a fair die, using a formula and a zero-based derivation.

Explore the hypo hypergeometric distribution for actuaries, with parameters m, k, n; derive its pdf and mean, and apply to urn problem counting red balls in six draws without replacement.

Explore the multinomial distribution with n trials and k outcomes, where counts x1...xk sum to n. Use a ten-roll die example to compute probability with factorial formula and p^x terms.

Compute the mode of the number of hurricanes required for two damages under a negative binomial model with r=2 and p=0.4. The lecture demonstrates that the mode is three hurricanes.

Derive the pmf for a nonnegative discrete random variable using p_{k-1} = (1/k) p_k to obtain p_n = p0/(n-1)!. Compute p0 from normalization, yielding p0 = 1/(1+e).

Apply geometric distribution to find the probability the sixth test is the first with high blood pressure, given E[X] = 12.5. Compute p = 1/12.5, then probability as (1-p)^5 p.

Compute the probability that two independent lives with a geometric distribution die in the same year by summing across years and applying the geometric series formula, yielding about 0.005025126.

Describe the continuous uniform distribution with parameters a and b, where pdf is 1/(b-a) on [a,b] and the CDF is (x-a)/(b-a). Show the mean (a+b)/2 and variance (b-a)^2/12.

Analyze the probability that a uniform variable on [a,b] is less than 1, across three cases: both under 1, crossing 1, or both over 1.

Explore the normal distribution with mu and sigma, the standard normal (mu=0, sigma=1), its pdf and cdf, and how mu and sigma determine the mean, variance, and graph shape.

Master reading the standard normal z-table to find probabilities for z with mu=0 and sigma=1, using the area to the left. See z=1.96 yielding 0.975 and symmetry about zero.

Standardize a normal variable using z = (x−μ)/σ and consult the z-table. The lecture computes P(X<4) for μ=3, σ=2 as 0.69146 and P(X<0.5) for μ=1, σ=1 as 0.30854, using symmetry.

Compute the 75th and 20th percentiles for a normal distribution using the standard normal and z-table. Derive mu on the original scale from those percentiles.

Solve for mu and sigma of a normal distribution using the 90th and 25th percentiles with z = (x - mu)/sigma and z-table values.

Learn how to apply the normal approximation to discrete probabilities and use integer correction to improve accuracy, with examples using thresholds like eight and seven.

Use normal approximation for Poisson(50) to estimate P(X ≤ 60). Compare results with and without integer correction, yielding 0.92135 and 0.93122, illustrating about a 1% difference.

Compare normal approximation with and without continuity correction for P(X<60) in a binomial with n=100 and p=0.7, giving about 0.011 and 0.014.

Apply the normal approximation to a binomial with n=40 and p=0.5 to estimate P(X≥25) with and without continuity correction.

the central limit theorem says the sum of iid random variables with any pdf converges to a normal distribution, and n at least 30 gives an approximate normal.

Explore the exponential distribution with parameter theta as a model for time until the next event, covering its pdf, cdf, survival function, and moments: mean theta and variance theta^2.

Solve an exponential distribution problem where P(X<2) = 2 P(X>4); derive e^{-2/θ} = 1/2, solve for θ ≈ 2.885, and compute variance θ² ≈ 8.325.

Compare hazard rate functions for a uniform[0,10] and an exponential(mean 10). Interpret the uniform hazard rising near 10 and the exponential hazard remaining constant, illustrating lack of memory property.

Explain how Poisson counts in a period imply exponential waiting times with consistent parameters, and illustrate the proof with a sample question.

Explore how Poisson arrivals with mean 12 in 24 hours yield exponential interarrival times with theta 2 hours, showing identical survival probabilities via Poisson and exponential analyses.

Learn how the minimum of independent exponential variables with means theta_i is exponential with mean theta_x, where theta_x equals the reciprocal of the sum of reciprocals of theta_i.

Apply the exponential distribution to actuarial problems: compute the expected refund for printers (mean two years) and solve for x to match 1000 in insurance payments (mean ten years).

Learn to use the exponential distribution to model time until the next hurricane, solve for theta from f(x)=1−e^{−x/θ}, and obtain an expected time of about 7.21.

Explore how the gamma distribution arises by summing alpha independent exponential variables with mean theta, yielding mean alpha theta and variance alpha theta^2.

model two independent normal errors, x ~ N(0,0.0056^2) and y ~ N(0,0.0044^2); determine the mean and variance of (x+y)/2 and the probability that it lies within ±0.005, about 0.8385.

Standardize with z = (x−μ)/σ to find s = 12 from the 80 and 90 scores, and compute E[x | x>0] for a normal pdf, yielding sqrt(2/pi).

derive the exponential distribution parameter from a four-hour median, then calculate the survival probability that a component lasts at least five hours, about 0.42.

Explore the joint, marginal, and conditional distributions of multiple random variables, with examples using two dice to illustrate independence, joint pdf, and probabilities such as p(x=1,y=5) and p(x=2,y=12).

Compute joint probabilities for discrete and continuous random variables using f(x,y), identify cases like x=1, y=1 and x<y, and visualize with two-dimensional plots.

Visualize joint distribution problems by drawing constraint regions and shading probabilities under joint density f(x,y), including x,y in [0,1], x in [0,2] with x+y<2, and events x>y, x>1, and y<0.5.

Learn how to set up double integration by squeezing the graph from left and right, choosing bounds a, b, c, d, and switching x and y to simplify the calculation.

Examine a double integral over the region between x = y and y = x + 1 for 0 ≤ x ≤ 1. Show dy-first and dx-first yield same result.

Set up and evaluate a double integral over the triangle with x≥0, y≥0, and x+y≤1, first with x then y, then show reversing the order gives the same result.

Plot the region bounded by x−y=1, x−y=−1, x+y=−1, and x+y=1, then set up a double integral for area with dx first and dy second, splitting into bottom and top halves.

Explain how to set up a double integral for the area bounded by x=0, y=0, y=2-2x, and x=y, showing that integrating with dy first is simpler than splitting for dx.

Set up the double integral for the region bounded by x in [0,2], y in [0,5] with line 3x=2y; illustrate both dx dy and dy dx bounds.

Calculate the expectation of a function h(x,y) for discrete and continuous cases using joint distributions, and practice from a 2x2 table to find E[X+Y], E[X], and E[X^2+Y^2].

Compute the expectation E[XY] for a discrete joint distribution by summing x y p(x,y) and discarding zero terms. Apply symmetry to the joint density f_{X,Y}=3/2(x^2+y^2) on the unit square to obtain P(X>Y)=1/2.

Compute the probability that x>1 for the joint density f(x,y)=0.75x over the triangle 0≤x≤2, 0≤y≤2−x, yielding about 0.46.

Compute P(Y<0.5) by integrating the joint density f(x,y) over the region y<0.5; compare two integration orders and use the easier x-first method to obtain 0.578125.

Compute the expectation of x via double integration, and set up joint density bounds for x^2+y^2=4 with x>0 to evaluate probabilities such as x>1 and y<0.

Learn the marginal distribution of x and y, focusing on one variable at a time. Use integration for continuous, and summation for discrete, to derive fx, px, fy, py.

Compute the marginal distributions for a discrete, jointly distributed pair by summing the joint probabilities over the other variable, and derive p_x and p_y from the table.

Derive the marginals from the joint pdf 0.75 x over the triangle 0≤x≤2, 0≤y≤2−x: f_X(x)=0.75x(2−x), 0≤x≤2; f_Y(y)=0.375(2−y)^2, 0≤y≤2.

Derives the marginal densities of a joint density f(x,y)=3/2(x^2+y^2) on the unit square, yielding f_x(x)=3/2 x^2+1/2 and f_y(y)=3/2 y^2+1/2, and highlighting their symmetry.

Examine independence of variables via if and only if, linking marginal to joint density and applying double integration to compute the probability that x is larger than y.

Derive the independence of x and y and express the joint density as f(x,y) = (1/200) e^{-x/10} e^{-y/20} for x>0, y>0. Calculate P(X>Y) by integrating over x>y and obtain 1/3.

Analyze independence of x and y via two cases: triangle x>0, y>0, x+y<2, and circle x^2+y^2=4, showing dependency when y’s range changes with x; independence implies a rectangle domain.

In the unit square, X and Y with joint density f(x,y) = (3/2)(x^2 + y^2) are not independent; the joint cannot equal the product of marginals, so they are dependent.

Analyze independence of two discrete random variables x and y with a given joint distribution, derive the marginals, and verify that f_XY equals f_X f_Y, confirming independence.

Learn to compute the expectation from the joint and marginal distributions using double integration, with f(x,y)=24xy on 0<x<1, y>0, x+y<1, and confirm E[X]=E[Y] for actuarial exam P.

Derive the marginal distributions from the joint density f(x,y)=2 e^{-(x+2y)} for x>0,y>0, yielding f_X(x)=e^{-x} and f_Y(y)=2 e^{-2y}, and compute the corresponding expectations.

Explore conditional probability using Bayes' theorem to compute P(X>1/2 | Y>1/2) from a joint density fX,Y on 0≤x≤1, 0≤y≤2, and practice setting up integrals.

Apply Bayes' theorem to calculate conditional probabilities, analyze x given y and y given x, and recognize independence in joint density examples.

Apply conditional probability and Bayes theorem to derive the joint and marginal densities, then compute the expected bonus for a loss variable with uniform and conditional distributions.

Apply conditional probability and Bayes' theorem to a car crash: compute the expected number of hospitalized people given total loss is less than one, with independent hospitalizations and uniform losses.

Master the double expectation rule by conditioning on Y to compute E[X] as E[E[X|Y]], illustrated with a coin toss and dice, and grasp the accompanying double variance rule.

Apply the double expectation rule to total loss S = sum of N exponential X_i, with N Poisson(4) and X_i mean 1000, yielding E[S] = 4000 and Var(S) = 8,000,000.

Use the double expectation rule on a random sum of exponential losses, with n ~ binomial(3, 0.25), to compute the mean and variance of the unreimbursed total 0.3 s.

Apply the double expectation rule to compute the expectation and variance of Poisson counts with random lambda uniformly distributed on [0,3]. Find E[n]=1.5 and Var(n)=2.25.

Analyze how covariance measures linear relationships via E[XY]-E[X]E[Y], and how correlation scales to -1 to 1, with variance rules for sums and linear combinations.

Compute the covariance and correlation for x and y under the given density on [0,1]². Derive E[xy], E[x], E[y], and variances to obtain the covariance of -1/64 and correlation -15/73.

Explore how to compute the covariance between x and y in a coin-toss and dice scenario, using expectations and case analysis to show a positive covariance.

Explore covariance and correlation for the joint distribution f(x,y)=2x on x in [0,1], y in [x,x+1], deriving E[XY], E[X], E[Y], the variances, covariance -7/36, and the correlation (approx -1.2).

Explore the bivariate moment generating function for a joint distribution; differentiate w.r.t. t1 and t2, evaluate at zero to obtain moments and the covariance of x and y.

Explore the transformation of variables through one-to-one mappings, derive the pdf and cdf of y from x using g inverse, and apply the chain rule in the transformation proof.

Examine a one-to-one transformation where y = t^2 and y>4, deriving the pdf of y from t's cdf using two methods: formula and logic.

Transform an exponential(1) variable x using y = 2 ln x to derive the pdf of y, applying both the formula method and the logic (cdf) method.

Explore when the transformation x = y^2 is 1-to-1, apply logic for non-injective ranges, and derive the pdf f_X(a) = e^{-√a}/(2√a) for a > 0.

derive the pdf of y from y=10 x^0.8 with x drawn from an exponential distribution with mean one, using the one-to-one transformation, via both the formula and a CDF-based approach.

Using the cdf, this lecture derives the distribution of a = x + y on a unit square; the pdf f_A(k) = k for 0 ≤ k ≤ 2.

Derive the pdf of the ratio a = x/y from the joint density e^{-x-y} for x>0 and y>0 by using the CDF approach, yielding f_A(a) = 1/(a+1)^2.

Explore order statistics for any random variable, including the pdf of the k-th order statistic, and apply it to the expected time until the second exponential flight arrives.

Explore how limits and deductibles shape insurance payments, using the loss X, the payment xl, and the deductible xd to illustrate discrete and continuous cases and key expectation formulas.