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Questions and Answers
What is the relationship between the random variables X and Y in the given context?
What is the relationship between the random variables X and Y in the given context?
The variable Y is related to X by the equation $Y = \frac{9X + 160}{5}$.
Calculate the mean of the temperature Y in Fahrenheit when the mean of X is 10 degrees Celsius.
Calculate the mean of the temperature Y in Fahrenheit when the mean of X is 10 degrees Celsius.
The mean of Y is 50 degrees Fahrenheit.
If the standard deviation of X is 10, what is the standard deviation of Y?
If the standard deviation of X is 10, what is the standard deviation of Y?
The standard deviation of Y is 18.
What is the probability that the temperature exceeds 77 degrees Fahrenheit?
What is the probability that the temperature exceeds 77 degrees Fahrenheit?
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Explain how to standardize the random variable X in this context.
Explain how to standardize the random variable X in this context.
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What transformation can be applied to convert a measurement in Celsius to Fahrenheit?
What transformation can be applied to convert a measurement in Celsius to Fahrenheit?
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What does the symbol Φ represent in the calculations for probabilities?
What does the symbol Φ represent in the calculations for probabilities?
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In the alternate solution, how is the standard deviation of Y calculated from X?
In the alternate solution, how is the standard deviation of Y calculated from X?
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What is the cumulative distribution function (CDF) 𝐹𝐹𝑍𝑍 (𝑧𝑧) for the distance from Bob's dart to the center of the target?
What is the cumulative distribution function (CDF) 𝐹𝐹𝑍𝑍 (𝑧𝑧) for the distance from Bob's dart to the center of the target?
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What is the probability density function (PDF) 𝑓𝑓𝑍𝑍 (𝑧𝑧) for the dart hitting the target?
What is the probability density function (PDF) 𝑓𝑓𝑍𝑍 (𝑧𝑧) for the dart hitting the target?
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How is the probability of the dart landing in a region related to the area's proportionality?
How is the probability of the dart landing in a region related to the area's proportionality?
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Calculate the mean value 𝐸[𝑍] for the distance of the dart from the center of the target.
Calculate the mean value 𝐸[𝑍] for the distance of the dart from the center of the target.
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What is the geometric interpretation of the CDF 𝐹𝐹𝑍𝑍 (𝑧𝑧) in the context of hitting a circular target?
What is the geometric interpretation of the CDF 𝐹𝐹𝑍𝑍 (𝑧𝑧) in the context of hitting a circular target?
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What does the value of 1 in the CDF indicate about the dart's hitting range?
What does the value of 1 in the CDF indicate about the dart's hitting range?
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Describe how to find the variance 𝑉𝑎𝑟(𝑍) based on the provided information.
Describe how to find the variance 𝑉𝑎𝑟(𝑍) based on the provided information.
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Why is the value of the PDF 𝑓𝑓𝑍𝑍 (𝑧𝑧) equal to 0 outside the interval [0, r]?
Why is the value of the PDF 𝑓𝑓𝑍𝑍 (𝑧𝑧) equal to 0 outside the interval [0, r]?
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What is the total expectation theorem equation given in the content?
What is the total expectation theorem equation given in the content?
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What is the probability density function (PDF) of the random variable $X$ given in the content?
What is the probability density function (PDF) of the random variable $X$ given in the content?
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Define the PDF of $Y$ conditioned on $X$ as provided.
Define the PDF of $Y$ conditioned on $X$ as provided.
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What is the integral used to compute the PDF of $Y$?
What is the integral used to compute the PDF of $Y$?
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After evaluating the integral, what is the final expression for $f_Y(y)$?
After evaluating the integral, what is the final expression for $f_Y(y)$?
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What is the value of $p$ for the uniform distribution of $X$?
What is the value of $p$ for the uniform distribution of $X$?
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Explain the significance of the variance of $Y$ being equal to 1 in this context.
Explain the significance of the variance of $Y$ being equal to 1 in this context.
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How does the expectation theorem relate to conditional expectations in this context?
How does the expectation theorem relate to conditional expectations in this context?
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What does $\Phi(z)$ represent in the expression for $f_Y(y)$?
What does $\Phi(z)$ represent in the expression for $f_Y(y)$?
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Why is it essential to specify $y \in ℝ$ when stating the PDF of $Y$?
Why is it essential to specify $y \in ℝ$ when stating the PDF of $Y$?
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What does the formula for 𝑓𝑓𝑍𝑍 (𝑧𝑧) represent in terms of the functions 𝑓𝑓𝑋𝑋 and 𝑓𝑓𝑌𝑌?
What does the formula for 𝑓𝑓𝑍𝑍 (𝑧𝑧) represent in terms of the functions 𝑓𝑓𝑋𝑋 and 𝑓𝑓𝑌𝑌?
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Under what conditions is the integrand non-zero?
Under what conditions is the integrand non-zero?
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What is the significance of the boundary conditions $a + c ≤ z ≤ b + d$?
What is the significance of the boundary conditions $a + c ≤ z ≤ b + d$?
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Explain why the interval of integration can become empty.
Explain why the interval of integration can become empty.
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How is 𝑓𝑓𝑍𝑍 (𝑧𝑧) calculated when the boundaries are satisfied?
How is 𝑓𝑓𝑍𝑍 (𝑧𝑧) calculated when the boundaries are satisfied?
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What happens to 𝑓𝑓𝑍𝑍 (𝑧𝑧) when the conditions are not met?
What happens to 𝑓𝑓𝑍𝑍 (𝑧𝑧) when the conditions are not met?
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Define the components $m(a, z−d)$ and $m(b, z−c)$ in the equation for 𝑓𝑓𝑍𝑍 (𝑧𝑧).
Define the components $m(a, z−d)$ and $m(b, z−c)$ in the equation for 𝑓𝑓𝑍𝑍 (𝑧𝑧).
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What mathematical operations are performed in calculating the value of 𝑓𝑓𝑍𝑍 (𝑧𝑧)?
What mathematical operations are performed in calculating the value of 𝑓𝑓𝑍𝑍 (𝑧𝑧)?
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What does the function 𝐹𝐹𝑍𝑍(𝑧𝑧) represent in the context of $Z = max(X, Y)$?
What does the function 𝐹𝐹𝑍𝑍(𝑧𝑧) represent in the context of $Z = max(X, Y)$?
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How does the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧) relate to the cases where $b < d$ and $d ≤ b$?
How does the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧) relate to the cases where $b < d$ and $d ≤ b$?
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What condition is given for the random variables $X$ and $Y$ in the problem?
What condition is given for the random variables $X$ and $Y$ in the problem?
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What formula is used to calculate the joint probability in terms of the individual distributions $F_X(z)$ and $F_Y(z)$?
What formula is used to calculate the joint probability in terms of the individual distributions $F_X(z)$ and $F_Y(z)$?
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In case 1 where $b < d$, what is the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧)$ in the interval $[b, d]$?
In case 1 where $b < d$, what is the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧)$ in the interval $[b, d]$?
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What is the value of the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧)$ when $z < max(a, c)$?
What is the value of the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧)$ when $z < max(a, c)$?
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What does the expression $(b - a)(d - c)$ represent in the case formula?
What does the expression $(b - a)(d - c)$ represent in the case formula?
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Can $Z = X + Y$ also be treated similarly to $Z = max(X, Y)$ in terms of finding its PDF?
Can $Z = X + Y$ also be treated similarly to $Z = max(X, Y)$ in terms of finding its PDF?
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How does the independence of $X$ and $Y$ simplify the computation of $F_Z(z)$?
How does the independence of $X$ and $Y$ simplify the computation of $F_Z(z)$?
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What is the implication of having $f_Z(z) = 0$ for certain ranges of $z$?
What is the implication of having $f_Z(z) = 0$ for certain ranges of $z$?
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What is the probability density function (PDF) for the random variable $X$ uniformly distributed on the interval $[a, b]$?
What is the probability density function (PDF) for the random variable $X$ uniformly distributed on the interval $[a, b]$?
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How do we express the cumulative distribution function (CDF) for the random variable $Y$ uniformly distributed on the interval $[c, d]$?
How do we express the cumulative distribution function (CDF) for the random variable $Y$ uniformly distributed on the interval $[c, d]$?
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Given independent random variables $X$ and $Y$, how would you define the PDF of the random variable $Z = \max(X, Y)$?
Given independent random variables $X$ and $Y$, how would you define the PDF of the random variable $Z = \max(X, Y)$?
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What are the properties of PDFs that apply to the functions $f_X(x)$ and $f_Y(y)$?
What are the properties of PDFs that apply to the functions $f_X(x)$ and $f_Y(y)$?
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Write down the expression for the CDF of $X$ for the case when $x$ is between $a$ and $b$.
Write down the expression for the CDF of $X$ for the case when $x$ is between $a$ and $b$.
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Explain the significance of the condition '$X$ and $Y$ are independent' in finding the joint PDF.
Explain the significance of the condition '$X$ and $Y$ are independent' in finding the joint PDF.
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Calculate the expected value of $Y$ if $y$ is uniformly distributed over $[c, d]$.
Calculate the expected value of $Y$ if $y$ is uniformly distributed over $[c, d]$.
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What does the integral of the PDF from negative infinity to positive infinity equal for $X$?
What does the integral of the PDF from negative infinity to positive infinity equal for $X$?
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How would you formulate the PDF for $Z$ when $Z = \max(X, Y)$, if $X$ and $Y$ are independent?
How would you formulate the PDF for $Z$ when $Z = \max(X, Y)$, if $X$ and $Y$ are independent?
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What are the values of $F_X(x)$ and $F_Y(y)$ when $x < a$ and $y < c$, respectively?
What are the values of $F_X(x)$ and $F_Y(y)$ when $x < a$ and $y < c$, respectively?
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Study Notes
Review Problems - Solutions
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Problem 1a: Find the CDF Fz(z) and PDF fz(z) for a dart thrown at a circular target of radius r. Bob hits the target with equal probability at any point. The probabilities are proportional to the areas. The CDF is 0 if z < 0, z² / r² if 0 ≤ z < r, and 1 if z > r. The PDF is 2z/r² if 0 ≤ z < r, and 0 otherwise.
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Problem 1b: Calculate the mean E[Z]. The mean is 2r/3.
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Problem 1c: Find the variance Var(Z). The variance is r²/18.
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Problem 2: A city's temperature in Celsius (X) follows a normal distribution with a mean of 10 and standard deviation of 10. Temperature in Fahrenheit (Y) is related to Celsius by X = 5(Y-32)/9. What is the probability that the temperature is above 77°F? The probability is 0.0668.
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Problem 3a: Determine the joint PDF fx,y(x, y). The region is a triangle with vertices (0, 1), (1, 0), and (1, 1). The joint PDF is 2 if 0 < y ≤ 1 and 1-y ≤ x ≤ 1, and 0 otherwise.
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Problem 3b: Determine the marginal PDF fx(x). The marginal PDF is 2x if 0 ≤ x ≤ 1, and 0 otherwise.
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Problem 3c: Determine the expected value E[X]. The expected value is 2/3.
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Problem 3d: Determine the conditional expectation E[X|Y = y]. The conditional expectation is (1 - (1 - y)²) / 2y.
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Problem 4: Let X be an exponential random variable with parameter λ, and Y = X + 1. Determine the PDF fy(y). The PDF is λe^(-λ(y-1)) if y ≥ 1, and 0 otherwise.
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Problem 5: Alice has a choice of two games at the casino. The first game's winnings (X) are normally distributed with μ = 1 and σ = 2; the second game's winnings are uniformly distributed between -1 and 2. What is her expected winnings? Expected winnings are given by the probability of choosing the first game times the expected winnings for that game plus the probability of choosing the second game times expected winnings for that.
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Problem 6a: Find the PDF fy(y) for a normal random variable Y with variance 1 and mean X, where X is uniformly distributed on the interval [1, 3].
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Problem 6b: Find the conditional PDF fx|y(x|y). This is determined using Bayes' rule: fx|y(x|y) = fy|x(y|x) fx(x)/fy(y).
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Problem 7a: Write the PDFs fx(x) and fy(y) for independent, jointly continuous random variables X and Y where X is uniform on [a, b] and Y is uniform on [c, d].
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Problem 7b: Write the CDFs Fx(x) and Fy(y). The CDFs are defined in terms of intervals.
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Problem 7c: Calculate the PDF fz(z) for Z = max(X, Y).
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Problem 7d: Calculate the PDF fz(z) for Z = X + Y. The PDF is determined based on the conditions on a, b, c, and d.
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Description
This quiz covers key concepts in probability and statistics, including cumulative distribution functions, probability density functions, and calculations of mean and variance. Additionally, the quiz investigates normal distribution and joint probability distributions in specific contexts. Test your knowledge and understanding of these essential statistical principles.