Question

Let Z(t)=Xt+Y, where X is a Gaussian random variable with mean 0 and variance 2, and Y is a Gaussian random variable with mean 1 and variance 1. Assume X and Y are independent. (a) Find the mean function m Z (t)

Answer 1

To calculate the z-score for y = 4 using Y's distribution, we use the formula z = (y - μ) / σ. Substituting in the given values, we find that z = (4 - 2) / 1 = 2.

Step-by-step explanation:

If the random variables X and Y have the normal distributions X ~ N(5, 6) and Y ~ N(2, 1), and it's given that z corresponds to x = 17 via X's distribution, we can find the corresponding z for y = 4 via Y's distribution. The z-score is calculated using the formula z = (y - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, substitute y = 4, μ = 2 (mean of Y), and σ = 1 (standard deviation of Y) to get the z-score for Y.

Therefore, z = (4 - 2) / 1 = 2.