Determine Positive Values for a and b Given Mean and Variance Conditions for Random Variable Transformations

In this article, we will explore how to find the positive values of parameters a and b such that a linear transformation of a given random variable X results in a new random variable Y with specified mean and variance. This involves understanding the properties of the mean and variance of linear transformations of random variables.

Understanding the Problem

We are given a continuous random variable X with a mean of 10 and a variance of 25. Our goal is to find positive values for a and b such that the transformed random variable Y aX b has a mean of 0 and a variance of 1.

Using Properties of Means and Variances

To solve this problem, we will use the properties of the mean and variance of linear transformations of random variables. Specifically, for a random variable X, a linear transformation Y aX b has:

Mean of Y

[ E[Y] E[aX b] aE[X] b ]

Variance of Y

[ text{Var}(Y) text{Var}(aX b) a^2 text{Var}(X) ]

Step-by-Step Solution

Given:

We need to find positive values for a and b such that the mean and variance conditions are satisfied.

Calculating the Mean of Y

Using the property of the mean:

[ E[Y] aE[X] b a cdot 10 b 0 ]

Rearranging for b:

[ b -10a tag{1} ]

Calculating the Variance of Y

Using the property of the variance:

[ text{Var}(Y) a^2 text{Var}(X) a^2 cdot 25 1 ]

Solving for a:

[ a^2 cdot 25 1 ]

[ a^2 frac{1}{25} ]

Using the positive value for a:

[ a frac{1}{5} ]

Substituting a into (1) to Find b

Using b -10a:

[ b -10 cdot frac{1}{5} -2 ]

Hence, the positive values of a and b that satisfy the conditions are:

[ a frac{1}{5} ]

[ b -2 ]

Conclusion

The positive values for a and b such that Y aX b has a mean of 0 and a variance of 1 are:

[ boxed{a frac{1}{5} quad b -2} ]

This solution is based on the properties of the mean and variance of linear transformations, ensuring that the transformed random variable meets the specified conditions.

Keywords: random variable, linear transformation, mean, variance, conditional transformation