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Questions and Answers
If $X$ follows a Poisson distribution, what parameter(s) completely define its probability mass function (p.m.f.)?
If $X$ follows a Poisson distribution, what parameter(s) completely define its probability mass function (p.m.f.)?
- The variance, $\sigma^2$
- The mean, $\mu$ (correct)
- The standard deviation, $\sigma$
- The sample size, $n$
Which of the following is a necessary condition for a transformation $y = u(x)$ to be invertible, allowing for the determination of $x = w(y)$?
Which of the following is a necessary condition for a transformation $y = u(x)$ to be invertible, allowing for the determination of $x = w(y)$?
- $u(x)$ must be strictly monotonic (either strictly increasing or strictly decreasing). (correct)
- $u(x)$ must be differentiable.
- $u(x)$ must be continuous.
- $u(x)$ must be a linear function.
When transforming a discrete random variable, the support of the transformed variable is always the same as the support of the original variable.
When transforming a discrete random variable, the support of the transformed variable is always the same as the support of the original variable.
False (B)
If $Y = X^2$ and $X$ can take values {0, 1, 2, 3}, what are the possible values that $Y$ can take?
If $Y = X^2$ and $X$ can take values {0, 1, 2, 3}, what are the possible values that $Y$ can take?
For a one-to-one transformation, the inverse transformation is denoted as $x = ______$.
For a one-to-one transformation, the inverse transformation is denoted as $x = ______$.
When finding the p.m.f. of $Y = u(X)$, where $X$ is a discrete random variable, which of the following steps is typically involved?
When finding the p.m.f. of $Y = u(X)$, where $X$ is a discrete random variable, which of the following steps is typically involved?
If a transformation $y = u(x)$ is not one-to-one, it is impossible to find the p.m.f. of the transformed variable $Y$.
If a transformation $y = u(x)$ is not one-to-one, it is impossible to find the p.m.f. of the transformed variable $Y$.
Given the transformation $Y = 4X$, if the support of $Y$ is {0, 4, 8,...}, what is the support of $X$?
Given the transformation $Y = 4X$, if the support of $Y$ is {0, 4, 8,...}, what is the support of $X$?
For a discrete random variable $X$, the set of all possible values that $X$ can take is called its ______.
For a discrete random variable $X$, the set of all possible values that $X$ can take is called its ______.
What is the primary purpose of transforming a random variable?
What is the primary purpose of transforming a random variable?
When dealing with bivariate transformations, the Jacobian determinant is used to find the joint p.m.f. of the transformed variables.
When dealing with bivariate transformations, the Jacobian determinant is used to find the joint p.m.f. of the transformed variables.
If $Y_1 = X_1X_2$ and $Y_2 = X_2$, express $X_1$ and $X_2$ in terms of $Y_1$ and $Y_2$.
If $Y_1 = X_1X_2$ and $Y_2 = X_2$, express $X_1$ and $X_2$ in terms of $Y_1$ and $Y_2$.
In the context of bivariate transformations, the functions that express the original variables in terms of the transformed variables are called ______.
In the context of bivariate transformations, the functions that express the original variables in terms of the transformed variables are called ______.
In bivariate transformations, what condition must be satisfied to ensure a proper transformation from $(X_1, X_2)$ to $(Y_1, Y_2)$?
In bivariate transformations, what condition must be satisfied to ensure a proper transformation from $(X_1, X_2)$ to $(Y_1, Y_2)$?
Match the transformation type with its corresponding mathematical operation:
Match the transformation type with its corresponding mathematical operation:
Define the term 'support' in the context of a discrete random variable.
Define the term 'support' in the context of a discrete random variable.
If $X$ has a binomial distribution and $Y = X/n$, what is the potential issue in directly applying the discrete transformation techniques?
If $X$ has a binomial distribution and $Y = X/n$, what is the potential issue in directly applying the discrete transformation techniques?
When transforming discrete random variables, it is always possible to find a closed-form expression for the p.m.f. of the transformed variable.
When transforming discrete random variables, it is always possible to find a closed-form expression for the p.m.f. of the transformed variable.
When a transformation is not one-to-one, to find $P(Y=y)$, one must sum the probabilities of all ______ values that map to $y$.
When a transformation is not one-to-one, to find $P(Y=y)$, one must sum the probabilities of all ______ values that map to $y$.
Consider a transformation $Y = u(X)$. Even if you know the p.m.f. of $X$ and the transformation $u$, what additional information is crucial to determine the p.m.f. of $Y$?
Consider a transformation $Y = u(X)$. Even if you know the p.m.f. of $X$ and the transformation $u$, what additional information is crucial to determine the p.m.f. of $Y$?
Flashcards
What is Probability Mass Function (PMF)?
What is Probability Mass Function (PMF)?
A function that gives the probability that a discrete random variable is exactly equal to some value.
What is a one-to-one transformation?
What is a one-to-one transformation?
A transformation where y = 4x, mapping A onto B, used to find the PMF of a new random variable.
What is the inverse transformation?
What is the inverse transformation?
Given a function Y = u(X), find X = w(y) to express the original variable in terms of the transformed variable.
What is a bivariate discrete transformation?
What is a bivariate discrete transformation?
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What are single-valued inverses?
What are single-valued inverses?
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What is a distribution technique?
What is a distribution technique?
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Study Notes
- Chapter 4 addresses transformations of discrete random variables
Transformation of Univariate Discrete Variables
- Given a random variable X with a known probability mass function (p.m.f.), a new random variable Y = u(X) can be defined.
- The goal is to find the p.m.f. of Y based on the transformation u.
Example 1
- X has a Poisson p.m.f. defined as f(x) = (μ^x * e^(-μ)) / x! for x = 0, 1, 2, ...
- Y is defined as Y = 4X.
- The transformation y = 4x maps set A to B, where A includes all possible varibles of X(0,1,2 ..) and B (y:y=0,4,8...).
- The p.m.f. of Y can be found using P(Y = y) = P(X = y/4) = (μ^(y/4) * e^(-μ)) / (y/4)! for y = 0, 4, 8, ...
Example 2
- X has a binomial p.m.f. P(X = x) = (3! / (x!(3-x)!)) * (2/3)^x * (1/3)^(3-x) for x = 0, 1, 2, 3
- Y is defined as Y = X².
- The transformation y = x² maps A to B, where A includes all possible varibles of X(0,1,2,3) and B (y:y=0,1,4,9).
- The p.m.f. of Y is P(Y = y) = P(X = √y) for y ∈ B, which translates to (3! / (√y!(3 - √y)!)) * (2/3)^√y * (1/3)^(3-√y).
Theorem
- If X has a p.m.f. f, and Y = u(X), where u is a one-to-one function on A = {x : f(x) > 0} mapping to B
- If x = w(y) is the inverse transformation, the p.m.f. of Y is g(y) = f(w(y)) for y ∈ B
Transformation of Bivariate Discrete Variables
- Discrete transformation for two random variables requires two functions.
Theorem:
- Given random variables X₁ and X₂ with a joint p.m.f. f(x₁, x₂), two new random variables Y₁ and Y₂ can be defined as: Y₁ = u₁(X₁, X₂) and Y₂ = u₂(X₁, X₂).
- The joint p.m.f. of Y₁ and Y₂ is g(y₁, y₂) = P(u₁(X₁, X₂) = y₁, u₂(X₁, X₂) = y₂).
- If the transformation (x₁, x₂) -> (y₁, y₂) is one-to-one, mapping A = {(x₁, x₂) : f(x₁, x₂) > 0} onto B, there exist functions w₁ and w₂ such that x₁ = w₁(y₁, y₂) and x₂ = w₂(y₁, y₂).
- x₁ and x₂ are the single-valued inverses of y₁ = u₁(x₁, x₂) and y₂ = u₂(x₁, x₂).
- The joint p.m.f. of Y₁ and Y₂ is g(y₁, y₂) = f(w₁(y₁, y₂), w₂(y₁, y₂)) for (y₁, y₂) ∈ B.
Example
- X₁ and X₂ have a joint p.m.f. f(x₁, x₂) = (x₁x₂) / 18 for x₁ = 1, 2 and x₂ = 1, 2, 3.
- Y₁ = X₁X₂ and Y₂ = X₂.
- The transformation is y₁ = x₁x₂ and y₂ = x₂.
- The inverse transformations are x₁ = y₁/y₂ and x₂ = y₂.
- A = {(x₁, x₂) : (1, 1)(1, 2)(1, 3)(2, 1)(2, 2)(2, 3)} onto B = {(y₁, y₂) : (1, 1)(2, 2)(3, 3)(2, 1)(4, 2)(6, 3)}
- The joint p.m.f. of Y₁ and Y₂ is g(y₁, y₂) = (y₁/y₂)y₂ / 18 = y₁ / 18 for (y₁, y₂) ∈ B.
- The marginal p.m.f. of Y₁ can be deduced from this joint p.m.f.
- For example, the process involves defining a convenient function Y₂ = u₂(X₁, X₂), finding the joint p.m.f. of Y₁ and Y₂, and then finding the marginal p.m.f. of Y₁.
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