Understanding the Expectation and Variance of Independent Random Variables with Normal Distribution

Introduction

In the realm of probability theory, the concepts of expectation and variance play crucial roles in understanding the behavior of random variables. This article delves into the specifics of independent random variables (X_1) and (X_2), each following a normal distribution with an expectation of 0 and variance of 2. Our focus narrows down to finding the expectation and variance of the linear transformation (U frac{1}{2}X_1 - X_2). We explore the mathematical properties and steps involved in this process to ensure a clear understanding of the underlying principles.

Step 1: Defining the Transformation

We start by defining a new random variable (U) as follows:(U frac{1}{2}X_1 - X_2)

Step 2: Distribution of U

The transformation can be understood through linear algebra. Specifically, we are interested in the distribution of (U). The expression (U frac{1}{2}X_1 - X_2) suggests a linear combination of independent normal random variables. For (X_1) and (X_2), each with a normal distribution (N(0, 2)), (U) will also follow a normal distribution. However, the exact distribution requires further calculation.

Step 3: Calculating the Expectation of U

The expectation of a random variable measures the central tendency. Given the linearity of expectation, we can calculate the expectation of (U) as follows:[ mathbb{E}[U] 0 ]

Hence, the expectation of (U) is 0, denoted as:(mathbb{E}[U] 0)

Step 4: Calculating the Variance of U

The variance of a random variable measures its dispersion or spread. Using the properties of variance, specifically that for independent random variables, we can find the variance of (U) as:[ mathsf{Var}(U) frac{5}{2} ]

Therefore, the variance of (U) is (frac{5}{2}), denoted as:(mathsf{Var}(U) frac{5}{2})

Step 5: Expectation of 2U and Variance of 2U

For the transformation (2U), we can use the linearity of expectation and the properties of variance to find its expectation and variance.

The expectation of (2U) is:[ mathbb{E}[2U] 0 ]

Similarly, the variance of (2U) is:[ mathsf{Var}(2U) 10 ]

Therefore, the expectation and variance of (2U) are:(mathbb{E}[2U] 0) and (mathsf{Var}(2U) 10).

Step 6: Distribution of U

From the earlier steps, we have established that (U) follows a normal distribution with expectation 0 and variance (frac{5}{2}). However, the provided content mentions that (U) can be related to a chi distribution. This is correct in the context of the central limit theorem where the sum of squared independent normal variables results in a chi-squared distribution. However, in this case, the transformation is linear and the distribution remains normal.

For a chi distribution, the mean is given by substituting (k 1), where (k) is the degrees of freedom. Since we are dealing with a linear combination, the mean and variance of (U) are as derived above and do not fit the chi distribution directly.

Conclusion

In conclusion, the expectation and variance of the random variable (U frac{1}{2}X_1 - X_2) are 0 and (frac{5}{2}), respectively. When considering the transformation to (2U), the expectation remains 0 and the variance becomes 10. This understanding is crucial for applications in probability and statistics, particularly in the analysis of linear transformations of normally distributed random variables.