Probability and Statistics Quiz
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Questions and Answers

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.

The mean of Y is 50 degrees Fahrenheit.

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?

<p>The probability is approximately 0.0668.</p> Signup and view all the answers

Explain how to standardize the random variable X in this context.

<p>To standardize X, subtract the mean (10) and divide by the standard deviation (10), giving the formula $Z = \frac{X - 10}{10}$.</p> Signup and view all the answers

What transformation can be applied to convert a measurement in Celsius to Fahrenheit?

<p>The transformation is given by $F = \frac{9C + 160}{5}$.</p> Signup and view all the answers

What does the symbol Φ represent in the calculations for probabilities?

<p>Φ represents the cumulative distribution function of the standard normal distribution.</p> Signup and view all the answers

In the alternate solution, how is the standard deviation of Y calculated from X?

<p>The standard deviation of Y is calculated as $(\frac{9}{5}) \cdot 10$.</p> Signup and view all the answers

What is the cumulative distribution function (CDF) 𝐹𝐹𝑍𝑍 (𝑧𝑧) for the distance from Bob's dart to the center of the target?

<p>𝐹𝐹𝑍𝑍 (𝑧𝑧) = 0 if $z &lt; 0$, $ rac{z^2}{r^2}$ if $0 ≤ z ≤ r$, and $1$ if $z &gt; r$.</p> Signup and view all the answers

What is the probability density function (PDF) 𝑓𝑓𝑍𝑍 (𝑧𝑧) for the dart hitting the target?

<p>𝑓𝑓𝑍𝑍 (𝑧𝑧) = $ rac{2z}{r^2}$ if $0 ≤ z ≤ r$ and $0$ otherwise.</p> Signup and view all the answers

How is the probability of the dart landing in a region related to the area's proportionality?

<p>The probability is proportional to the area of the region within the target, given by the ratio of the area of that region to the total area of the target.</p> Signup and view all the answers

Calculate the mean value 𝐸[𝑍] for the distance of the dart from the center of the target.

<p>𝐸[𝑍] = $ rac{2}{3} r$.</p> Signup and view all the answers

What is the geometric interpretation of the CDF 𝐹𝐹𝑍𝑍 (𝑧𝑧) in the context of hitting a circular target?

<p>The CDF represents the probability that the dart lands within a circle of radius $z$, centered at the origin.</p> Signup and view all the answers

What does the value of 1 in the CDF indicate about the dart's hitting range?

<p>It indicates that there is a 100% probability that the dart lands within the entire target (radius $r$).</p> Signup and view all the answers

Describe how to find the variance 𝑉𝑎𝑟(𝑍) based on the provided information.

<p>The variance can be calculated using $V[Z] = E[Z^2] - (E[Z])^2$, where $E[Z^2]$ can be derived from the PDF.</p> Signup and view all the answers

Why is the value of the PDF 𝑓𝑓𝑍𝑍 (𝑧𝑧) equal to 0 outside the interval [0, r]?

<p>This is because the dart cannot land at distances greater than the radius of the target, hence probability is 0.</p> Signup and view all the answers

What is the total expectation theorem equation given in the content?

<p>The total expectation theorem is given by $E[X] = pE[X|A] + (1-p)E[X|A^c]$.</p> Signup and view all the answers

What is the probability density function (PDF) of the random variable $X$ given in the content?

<p>The PDF of $X$ is $f_X(x) = 1/2$ for $1 ext{ ≤ } x ext{ ≤ } 3$ and $0$ otherwise.</p> Signup and view all the answers

Define the PDF of $Y$ conditioned on $X$ as provided.

<p>The conditional PDF is $f_{Y|X}(y|x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(y-x)^2}{2}}$ for $1 ≤ x ≤ 3$ and $y ∈ ℝ$.</p> Signup and view all the answers

What is the integral used to compute the PDF of $Y$?

<p>The integral is $f_Y(y) = \int_{-\infty}^{\infty} f_{Y|X}(y|x) f_X(x) dx$.</p> Signup and view all the answers

After evaluating the integral, what is the final expression for $f_Y(y)$?

<p>The final expression for $f_Y(y)$ is $\frac{1}{2}(\Phi(3-y) - \Phi(1-y))$.</p> Signup and view all the answers

What is the value of $p$ for the uniform distribution of $X$?

<p>The value of $p$ is $1/2$ since the uniform distribution spans an interval of length 2.</p> Signup and view all the answers

Explain the significance of the variance of $Y$ being equal to 1 in this context.

<p>A variance of 1 indicates a standard normal distribution, influencing the shape of the PDF of $Y$.</p> Signup and view all the answers

How does the expectation theorem relate to conditional expectations in this context?

<p>The expectation theorem uses conditional expectations to express the overall expected value based on probabilities.</p> Signup and view all the answers

What does $\Phi(z)$ represent in the expression for $f_Y(y)$?

<p>$\Phi(z)$ represents the cumulative distribution function (CDF) of the standard normal distribution.</p> Signup and view all the answers

Why is it essential to specify $y \in ℝ$ when stating the PDF of $Y$?

<p>Specifying $y \in ℝ$ ensures that the PDF is defined for all real numbers and clarifies the domain.</p> Signup and view all the answers

What does the formula for 𝑓𝑓𝑍𝑍 (𝑧𝑧) represent in terms of the functions 𝑓𝑓𝑋𝑋 and 𝑓𝑓𝑌𝑌?

<p>The formula represents the convolution of the functions 𝑓𝑓𝑋𝑋 and 𝑓𝑓𝑌𝑌, integrated over the appropriate limits.</p> Signup and view all the answers

Under what conditions is the integrand non-zero?

<p>The integrand is non-zero if both $a ≤ x ≤ b$ and $c ≤ z − x ≤ d$ hold true.</p> Signup and view all the answers

What is the significance of the boundary conditions $a + c ≤ z ≤ b + d$?

<p>These boundary conditions determine when the interval of integration is non-empty, allowing for the calculation of 𝑓𝑓𝑍𝑍 (𝑧𝑧).</p> Signup and view all the answers

Explain why the interval of integration can become empty.

<p>The interval becomes empty if either $b &lt; z − d$ or $z − c &lt; a$ is satisfied.</p> Signup and view all the answers

How is 𝑓𝑓𝑍𝑍 (𝑧𝑧) calculated when the boundaries are satisfied?

<p>When the boundaries are satisfied, it is calculated as the difference between $m(b, z−c)$ and $m(a, z−d)$, divided by the product of $(b−a)(d−c)$.</p> Signup and view all the answers

What happens to 𝑓𝑓𝑍𝑍 (𝑧𝑧) when the conditions are not met?

<p>If the conditions are not met, 𝑓𝑓𝑍𝑍 (𝑧𝑧) equals zero.</p> Signup and view all the answers

Define the components $m(a, z−d)$ and $m(b, z−c)$ in the equation for 𝑓𝑓𝑍𝑍 (𝑧𝑧).

<p>$m(a, z−d)$ and $m(b, z−c)$ represent the cumulative distribution values at those specific points based on the given a, b, c, and d.</p> Signup and view all the answers

What mathematical operations are performed in calculating the value of 𝑓𝑓𝑍𝑍 (𝑧𝑧)?

<p>The calculation involves convolution, integration, and normalization of the resulting values.</p> Signup and view all the answers

What does the function 𝐹𝐹𝑍𝑍(𝑧𝑧) represent in the context of $Z = max(X, Y)$?

<p>The function 𝐹𝐹𝑍𝑍(𝑧𝑧) represents the cumulative distribution function of the random variable $Z$, which gives the probability that $Z$ is less than or equal to $z$.</p> Signup and view all the answers

How does the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧) relate to the cases where $b < d$ and $d ≤ b$?

<p>The PDF 𝑓𝑓𝑍𝑍(𝑧𝑧) differs in form for the two cases, reflecting the overlapping behavior and intervals defined by the bounds of $X$ and $Y$.</p> Signup and view all the answers

What condition is given for the random variables $X$ and $Y$ in the problem?

<p>The condition is that $a ext{ (lower bound of X)} ext{ } extless d ext{ (upper bound of Y)}$ and $c ext{ (lower bound of Y)} ext{ } extless b ext{ (upper bound of X)}$, ensuring the intervals overlap.</p> Signup and view all the answers

What formula is used to calculate the joint probability in terms of the individual distributions $F_X(z)$ and $F_Y(z)$?

<p>The formula used is $F_Z(z) = F_X(z) imes F_Y(z)$, leveraging the independence of $X$ and $Y$.</p> Signup and view all the answers

In case 1 where $b < d$, what is the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧)$ in the interval $[b, d]$?

<p>In the interval $[b, d]$, the PDF is $f_Z(z) = rac{1}{d - c}$.</p> Signup and view all the answers

What is the value of the PDF 𝑓𝑓𝑍𝑍(𝑧𝑧)$ when $z < max(a, c)$?

<p>The value of the PDF $f_Z(z) = 0$ when $z &lt; max(a, c)$.</p> Signup and view all the answers

What does the expression $(b - a)(d - c)$ represent in the case formula?

<p>$(b - a)(d - c)$ represents the area of the joint distribution for the overlapping intervals of random variables $X$ and $Y$.</p> Signup and view all the answers

Can $Z = X + Y$ also be treated similarly to $Z = max(X, Y)$ in terms of finding its PDF?

<p>Yes, $Z = X + Y$ can also be treated using convolution techniques to find its PDF.</p> Signup and view all the answers

How does the independence of $X$ and $Y$ simplify the computation of $F_Z(z)$?

<p>Independence allows us to multiply the individual probabilities to determine the cumulative distribution, greatly simplifying computations.</p> Signup and view all the answers

What is the implication of having $f_Z(z) = 0$ for certain ranges of $z$?

<p>It implies that the random variable $Z$ cannot take values in those ranges due to the constraints imposed by $X$ and $Y$’s intervals.</p> Signup and view all the answers

What is the probability density function (PDF) for the random variable $X$ uniformly distributed on the interval $[a, b]$?

<p>(f_X(x) = \frac{1}{b - a}) if (a \leq x \leq b), and (0) otherwise.</p> Signup and view all the answers

How do we express the cumulative distribution function (CDF) for the random variable $Y$ uniformly distributed on the interval $[c, d]$?

<p>(F_Y(y) = 0) if (y &lt; c), (F_Y(y) = \frac{y - c}{d - c}) if (c \leq y \leq d), and (1) if (y &gt; d).</p> Signup and view all the answers

Given independent random variables $X$ and $Y$, how would you define the PDF of the random variable $Z = \max(X, Y)$?

<p>The PDF of (Z) would be calculated using the joint distribution of (X) and (Y), factoring in their independence.</p> Signup and view all the answers

What are the properties of PDFs that apply to the functions $f_X(x)$ and $f_Y(y)$?

<p>Both PDFs must satisfy: (f_X(x) \geq 0) and (f_Y(y) \geq 0) for all (x) and (y), and their integrals over their respective ranges equal to 1.</p> Signup and view all the answers

Write down the expression for the CDF of $X$ for the case when $x$ is between $a$ and $b$.

<p>(F_X(x) = \frac{x - a}{b - a}) if (a \leq x \leq b).</p> Signup and view all the answers

Explain the significance of the condition '$X$ and $Y$ are independent' in finding the joint PDF.

<p>Independence implies that the joint PDF can be expressed as the product of the individual PDFs: (f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)).</p> Signup and view all the answers

Calculate the expected value of $Y$ if $y$ is uniformly distributed over $[c, d]$.

<p>(E[Y] = \frac{c + d}{2}).</p> Signup and view all the answers

What does the integral of the PDF from negative infinity to positive infinity equal for $X$?

<p>(\int_{-\infty}^{\infty} f_X(x) , dx = 1).</p> Signup and view all the answers

How would you formulate the PDF for $Z$ when $Z = \max(X, Y)$, if $X$ and $Y$ are independent?

<p>(f_Z(z) = f_X(z)F_Y(z) + f_Y(z)F_X(z)).</p> Signup and view all the answers

What are the values of $F_X(x)$ and $F_Y(y)$ when $x < a$ and $y < c$, respectively?

<p>(F_X(x) = 0) and (F_Y(y) = 0).</p> Signup and view all the answers

Flashcards

CDF of Z (FZ(z))

Cumulative Distribution Function of Z, calculates the probability that the distance from the center of the target is less than or equal to z.

PDF of Z (fZ(z))

Probability Density Function of Z, represents the probability of finding a dart at a particular distance z from the target's center.

Distance from target center (Z)

The distance between the point where the dart lands and the center of the circular target.

Mean of Z (E[Z])

Expected value of the random variable representing the distance Z. The average distance of dart throws to the center.

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Variance of Z (Var(Z))

Measures the dispersion (spread) of distances from the center of the target, quantifying how far on average the dart's distance from the center deviates from the mean.

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Circular Target Radius (r)

The radius of the circular target.

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Equally Likely Hits

The probability of hitting any point within the target is proportional to the area of that region. A uniform distribution

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Probability proportional to area

A probability statement based on target area proportion, critical part of calculation

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Temperature Conversion (Celsius to Fahrenheit)

The formula to convert temperature from degrees Celsius (X) to degrees Fahrenheit (Y): Y = (9X + 160)/5

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Normal Random Variable (Temperature)

A temperature (in Celsius or Fahrenheit) that follows a normal distribution.

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Mean of Temperature (Celsius)

The average temperature of a city in degrees Celsius represented by the variable X, having a value of 10.

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Standard Deviation of Temperature (Celsius)

The measure of the variability or dispersion in temperature data in Celsius. The value is 10.

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Mean of Temperature (Fahrenheit)

The average temperature in degrees Fahrenheit (Y) is equal to 50.

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Standard Deviation of Temperature (Fahrenheit)

Calculated as (9/5) * standard deviation of Celsius temp; in this case, 18.

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Probability (Temperature > 77°F)

The probability that the temperature is above 77 degrees Fahrenheit (equivalent to finding P(X > 25°C)).

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Finding Probability (using Z-scores)

The probability is found by standardizing the value using z-scores to find the area under the normal curve from 1.5 and above of the Z-score.

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Total Expectation Theorem

A theorem that calculates the expected value of a random variable by conditioning on another event or variable.

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Conditional Expectation

The expected value of a random variable given the occurrence of another event.

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Continuous Uniform Distribution

A probability distribution where all intervals of equal length within the range of the random variable have the same probability.

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Normal Random Variable

A random variable whose probability distribution is a normal distribution.

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Variance

A measure of the spread or dispersion of a probability distribution.

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PDF

Probability Density Function

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Conditional PDF

Probability Density Function of a random variable, given the value of another.

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Expected Value

The average value of a random variable.

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Probability Density

A measure of how likely a continuous random variable is to take on a specific value.

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Gaussian Distribution

A special case of a continuous probability distribution, often referred to as the normal distribution.

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Z = max(X, Y)

The random variable Z is defined as the maximum value between two independent random variables X and Y.

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Case 1: 𝑏𝑏 < 𝑑𝑑

The scenario where the upper bound of the first random variable's distribution (b) is less than the upper bound of the second random variable's distribution (d).

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Case 2: 𝑑𝑑 ≤ 𝑏𝑏

The scenario where the upper bound of the first random variable's distribution (d) is less than or equal to the upper bound of the second random variable's distribution (b).

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Independence of X and Y

The probability of one random variable (X) occurring has no influence on the probability of the other random variable (Y) occurring.

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Joint Probability (X ≤ z, Y ≤ z)

The probability that both random variables X and Y are less than or equal to a specific value z.

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What does 'uniform on [a, b]' mean?

It means that the random variable has an equal probability of taking on any value within the interval [a, b].

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How is the PDF of Z derived?

It is derived by considering the probability of Z being less than or equal to 'z' based on the PDFs of X and Y, and then differentiating the resulting CDF.

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Jointly Continuous

X and Y are jointly continuous random variables, meaning their combined behavior can be described by a continuous probability distribution.

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Z interval

Range of possible values for Z, where the integrand is non-zero, implying a probability greater than 0.

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fZ(z) equation

Mathematical expression for the PDF of Z, representing the probability density at a distance 'z' from the target center.

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Integration interval

Defines the range of values of 'x' where the integral is calculated, based on the conditions for non-zero integrand.

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Z = distance from target center

The distance between the point where the dart lands and the center of the circular target, represented by the random variable Z.

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Non-empty integration interval

Condition where the interval of integration has a length greater than zero, meaning there's a chance of finding a dart in that range.

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Empty integration interval

Condition where the interval of integration is empty (zero length), implying no chance of landing a dart in that range.

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Study Notes

Review Problems - Solutions

  • 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.

  • Problem 1b: Calculate the mean E[Z]. The mean is 2r/3.

  • Problem 1c: Find the variance Var(Z). The variance is r²/18.

  • 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.

  • 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.

  • Problem 3b: Determine the marginal PDF fx(x). The marginal PDF is 2x if 0 ≤ x ≤ 1, and 0 otherwise.

  • Problem 3c: Determine the expected value E[X]. The expected value is 2/3.

  • Problem 3d: Determine the conditional expectation E[X|Y = y]. The conditional expectation is (1 - (1 - y)²) / 2y.

  • 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.

  • 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.

  • 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].

  • 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).

  • 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].

  • Problem 7b: Write the CDFs Fx(x) and Fy(y). The CDFs are defined in terms of intervals.

  • Problem 7c: Calculate the PDF fz(z) for Z = max(X, Y).

  • 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.

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