Question

Let X have a Burr distribution with parameters α = 1, γ = 2, and θ = √1000, and let Y have a Pareto distribution with parameters α = 1 and θ = 1000. Let Z be a mixture of X and Y with equal weight on each component. Let W = _________.

Answer 1

The random variable W can be defined as the maximum of Z and 1000, and it follows a distribution defined by the Burr(1, 2, √1000) ∨ Pareto(1, 1000) distribution.

Step-by-step explanation:

The random variable W can be defined as the maximum of Z and 1000. Since Z is a mixture of two random variables X and Y with equal weights, we can find the distribution of Z by taking the maximum of the distributions of X and Y. The distribution of X is Burr(1, 2, √1000) and the distribution of Y is Pareto(1, 1000). So, the distribution of Z is Burr(1, 2, √1000) ∨ Pareto(1, 1000).

To find the distribution of the maximum, we need to find the cumulative distribution function (CDF) of Z. The CDF of Z can be found by taking the maximum of the CDFs of X and Y. Finally, we can find the probability density function (PDF) of Z by differentiating the CDF of Z.

Therefore, W = max(Z, 1000) follows a distribution defined by the Burr(1, 2, √1000) ∨ Pareto(1, 1000) distribution.

Answer 2

W is a mixture of the Burr distribution with parameters
`\( \alpha = 1 \), \( \gamma = 2 \), and \( \theta = √(1000) \),` and the Pareto distribution with parameters
`\( \alpha = 1 \) and \( \theta = 1000 \)`, where each component contributes equally to the mixture.

To define the random variable W as a mixture of two distributions X and Y with equal weight, we need to understand the properties of the distributions involved and then combine them appropriately.

1. Burr Distribution (X): The Burr distribution is a flexible distribution for modeling data with a wide range of shapes. It's characterized by parameters
`\( \alpha \), \( \gamma \), and \( \theta \). In this case, \( \alpha = 1 \), \( \gamma = 2 \), and \( \theta = √(1000) \).`

2. Pareto Distribution (Y): The Pareto distribution is a power-law distribution used in various social, scientific, geophysical, and many other phenomena. It's characterized by a scale parameter
`\( \theta \) and a shape parameter \( \alpha \). Here, \( \alpha = 1 \) and \( \theta = 1000 \).`

3. Mixture Distribution (Z): A mixture distribution combines two or more distributions where each component distribution contributes to the overall mixture with a certain weight. In this case, Z is a mixture of X and Y with equal weight on each component. This means each distribution contributes 50% to the mixture.

Defining W:

Let W be a random variable that follows the distribution Z . The probability density function (pdf) or cumulative distribution function cdf of W ,
`\( f_W(w) \),`can be expressed as a weighted average of the pdfs or cdfs of X and Y . If
`\( f_X(x) \) is the pdf of \( X \) and \( f_Y(y) \) is the pdf of \( Y \)`, then the pdf of W is:

`\[ f_W(w) = 0.5 \cdot f_X(w) + 0.5 \cdot f_Y(w) \]`

Similarly, if
`\( F_X(x) \) and \( F_Y(y) \)` are the cdfs of X and Y respectively, then the cdf of W is:

`\[ F_W(w) = 0.5 \cdot F_X(w) + 0.5 \cdot F_Y(w) \]`