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Let $X$ and $Y$ be independent random variables, each with a distribution that is $N(0,1)$. Let $Z=X+Y$. Find the integral that represents the cdf $G(z)=$ $P(X+Y \leq z)$ of $Z .$ Determine the pdf of $Z$. Hint: We have that $G(z)=\int_{-\infty}^{\infty} H(x, z) d x$, where $$ H(x, z)=\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp \left[-\left(x^{2}+y^{2}\right) / 2\right] d y $$ Find $G^{\prime}(z)$ by evaluating $\int_{-\infty}^{\infty}[\partial H(x, z) / \partial z] d x$.

Let $X$ and $Y$ be independent random variables, each with a distribution that is $N(0,1)$. Let $Z=X+Y$. Find the integral that represents the cdf $G(z)=$ $P(X+Y \leq z)$ of $Z .$ Determine the pdf of $Z$. Hint: We have that $G(z)=\int_{-\infty}^{\infty} H(x, z) d x$, where $$ H(x, z)=\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp \left[-\left(x^{2}+y^{2}\right) / 2\right] d y $$ Find $G^{\prime}(z)$ by evaluating $\int_{-\infty}^{\infty}[\partial H(x, z) / \partial z] d x$.

Introduction to Mathematical Statistics

Some Special Distributions

The Normal Distribution
Insurance Reimbursement An insurance policy reimburses a loss up to a benefit limit of $10 .$ The policyholder's loss, $Y,$ follows a distribution with density function: $$f(y)=\left\{\begin{array}{ll}{\frac{2}{y^{3}}} & {\text { for } y>1} \\ {0} & {\text { otherwise }}\end{array}\right.$$ What is the expected value of the benefit paid under the insurance policy? Choose one of the following. (Hint: The benefit paid will be equal to the actual loss if the actual loss is less than the limit. Otherwise it will equal the limit.) Source: Society of Actuaries. $\begin{array}{lllllll}{\text { (a) } 1.0} & {\text { (b) } 1.3} & {\text { (c) } 1.8} & {\text { (d) } 1.9} & {\text { (e) } 2.0}\end{array}$

Insurance Reimbursement An insurance policy reimburses a loss up to a benefit limit of $10 .$ The policyholder's loss, $Y,$ follows a distribution with density function: $$f(y)=\left\{\begin{array}{ll}{\frac{2}{y^{3}}} & {\text { for } y>1} \\ {0} & {\text { otherwise }}\end{array}\right.$$ What is the expected value of the benefit paid under the insurance policy? Choose one of the following. (Hint: The benefit paid will be equal to the actual loss if the actual loss is less than the limit. Otherwise it will equal the limit.) Source: Society of Actuaries. $\begin{array}{lllllll}{\text { (a) } 1.0} & {\text { (b) } 1.3} & {\text { (c) } 1.8} & {\text { (d) } 1.9} & {\text { (e) } 2.0}\end{array}$

Calculus with Applications

Probability and Calculus

Expected Value and Variance of…
Insurance Reimbursement An insurance policy reimburses a loss up to a benefit limit of $10 .$ The policyholder's loss, $Y$ follows a distribution with density function: $$ f(y)=\left\{\begin{array}{ll}{\frac{2}{y^{3}}} & {\text { for } y>1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ What is the expected value of the benefit paid under the insurance policy? Choose one of the following. (Hint: The benefit paid will be equal to the actual loss if the actual loss is less than the limit. Otherwise it will equal the limit.) Source: Society of Actuaries. $\begin{array}{llllll}{\text { a. } 1.0} & {\text { b. } 1.3} & {\text { c. } 1.8} & {\text { d. } 1.9} & {\text { e. } 2.0}\end{array}$

Insurance Reimbursement An insurance policy reimburses a loss up to a benefit limit of $10 .$ The policyholder's loss, $Y$ follows a distribution with density function: $$ f(y)=\left\{\begin{array}{ll}{\frac{2}{y^{3}}} & {\text { for } y>1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ What is the expected value of the benefit paid under the insurance policy? Choose one of the following. (Hint: The benefit paid will be equal to the actual loss if the actual loss is less than the limit. Otherwise it will equal the limit.) Source: Society of Actuaries. $\begin{array}{llllll}{\text { a. } 1.0} & {\text { b. } 1.3} & {\text { c. } 1.8} & {\text { d. } 1.9} & {\text { e. } 2.0}\end{array}$

Finite Mathematics and Calculus with Applications

Probability and Calculus

Expected Value and Variance of…
Suppose that $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from an exponentially distributed population with mean $\theta$. Find the MLE of the population variance $\theta^{2}$. [Hint: Recall Example 9.9.]

Suppose that $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from an exponentially distributed population with mean $\theta$. Find the MLE of the population variance $\theta^{2}$. [Hint: Recall Example 9.9.]

Mathematical Statistics with Applications

Properties of Point Estimators and…

The Method of Maximum Likelihood

Questions asked

INSTANT ANSWER

Losses have a Pareto distribution with alpha = 0.5 and theta = 10,000. Determine the mean excess loss at 10,000.

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ANSWERED

Hamzah Choudary verified

Numerade educator

Claims have a Pareto distribution with alpha = 2 and theta unknown. Claims the following year experience 6% uniform inflation. Let r be the ratio of the proportion of claims that will exceed d next year to the proportion of claims that exceed d this year. Determine the limit of r as d goes to infinity.

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ANSWERED

Willis James verified

Numerade educator

Losses have a Pareto distribution with alpha = 0.5 and theta = 10,000. Determine the mean excess loss at 10,000.

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ANSWERED

Hamzah Choudary verified

Numerade educator

Losses have a Pareto distribution with alpha = 0.5 and theta = 10,000. Determine the mean excess loss at 10,000.

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INSTANT ANSWER

Negate (p->q)->q

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INSTANT ANSWER

building is worth @200,000 just before a loss. It is insured by a policy that has an X% coinsurance clause. The amount of the loss is $10,00C and the insured carried $120,000 of ccverage. The insurer pays $7500. Calculate X 2. In each of the following cases, what will the insurer pay or a claim of $12,000? 1. A 20% deductible and a policy limit of $ 12,500. 2. A straight deductible of $1000 and a policy limit of $10,000. 3. A linearly disappearing deductible such that a claim of $5000 has no loss payment but a claim of $15,000 is paid in full.

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ANSWERED

Akash M verified

Numerade educator

A market gardener faces the possibility of an early frost that would destroy part of the crop. The gardener can buy crop insurance. This creates four possible outcomes for the gardener's profit: Freeze | No Freeze No Insurance: 10,000 | 30,000 Insurance: 20,000 | 25,000 a) Based on expected monetary value, what probability would the farmer have to attach to early frost in order to make buying insurance a rational decision? (b) Given existing wealth, the farmer has the following utility profile: Profit | Utility 10,000 | 71 20,000 | 123 25,000 | 141 30,000 | 158 Based on expected utility, what probability would the farmer have to attach to early frost in order to make buying insurance a rational decision?

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ANSWERED

Jacob Fry verified

Numerade educator

Let Y1, ..., Yn be a random sample from a distribution with pdf f(y) = θy^(θ−1) where θ > 0 and 0 < y < 1. a) Find E(Y). b) Find the method of moments estimator for θ. c) Let X¯ be an estimator of θ. Is it an unbruised estimator? show work

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ANSWERED

Jacob Fry verified

Numerade educator

Suppose that the joint pdf of the random variables Y1 and Y2 is given by f(y1, y2) = 6y^2 * y2 if 0 < y1 < 1 and 0 < y2 < 1, and f(y1, y2) = 0, otherwise. a) Find the marginal pdf of Y1. Include the support. b) Find E(Y1). c) Find V (Y1). d) Are Y1 and Y2 independent? Explain

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ANSWERED

Dominador Tan verified

Numerade educator

Let Y1, ..., Yn be a random sample from a distribution with pdf f(y) = (α θ^α)/(y^α+1) where 0 < θ ≤ y, θ is known, and α > 0. a) Find the maximum likelihood estimator of α. (Make sure that you prove that your answer is the MLE.)

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