Final answer:
The expected value of the product of the sample mean and the average of a random variable, E(aˉTˉ), when μ=0.03 and δ=0.09, is 0.0009.
Explanation:
To calculate E(aˉTˉ), we use the property that the expected value of a product is the product of the expected values when dealing with independent variables. The formula for E(aˉTˉ) is E(aˉ) * E(Tˉ). The expected value of the sample mean (aˉ) is μ=0.03, and the expected value of the average of a random variable (Tˉ) is also μ=0.03. Multiplying these values together gives us 0.03 * 0.03 = 0.0009, which is the expected value of aˉTˉ.
In this context, the calculation involves finding the expected value of the product of two random variables. By definition, the expected value of a product of independent random variables is the product of their individual expected values. Here, aˉ represents the sample mean and Tˉ denotes the average of a random variable. With both having an expected value of μ=0.03, multiplying these expected values provides the final result of 0.0009 for E(aˉTˉ).
The outcome showcases the expectancy when these variables are combined, emphasizing how the expected value of their product relates directly to the individual expected values. This calculation is vital in understanding the expected behavior or central tendencies when dealing with two independent random variables and their combined effects. Therefore, with known individual expected values, the computation of the expected value of their product enables better predictive insights into their joint behavior.