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Consider the following independent random variables: U - X² (2) and V - X² (5). What is the

distribution of U+V? What is the distribution of (U/2)/(V/5)?

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Final answer:

The distribution of U + V is determined by adding the individual distributions of U and V, which are both dependent on the random variable X following a uniform distribution U(0,1). The distribution of (U/2)/(V/5) is obtained by dividing the individual distributions of U/2 and V/5, also based on the uniform distribution U(0,1).

Step-by-step explanation:

The random variables U and V are defined as U = X² (2) and V = X² (5), where X is a random variable following a uniform distribution U(0,1). To find the distribution of U + V, we need to determine the distribution of the sum of two independent random variables.

The sum of independent random variables is equal to the sum of their individual distributions. Since U and V are both dependent on X, we can rewrite U + V as (X² (2)) + (X² (5)). By substituting X with its distribution U(0,1), we get (U(0,1)² (2)) + (U(0,1)² (5)).

To obtain the final distribution, we can use the properties of the uniform distribution. The mean (μ) of X ~ U(0,1) is 0.5, and the variance (σ²) is 1/12. By substituting these values into the formula for variance, we can find the new mean and variance for U + V.

Regarding the distribution of (U/2)/(V/5), we can apply the concept of ratios of independent random variables. The ratio of independent random variables is equal to the ratio of their individual distributions. By dividing U/2 and V/5, we get (X²/2) and (X²/5), respectively. Again, substituting X with U(0,1), we have (U(0,1)²/2) and (U(0,1)²/5). We can compute the mean and variance of these new distributions to further analyze their properties.

User Premkumar Agrawal
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