Final answer:
The question requires knowledge of expected values, variance, and z-scores for random variables in normal distributions. To compute expected values and variances for transformed variables, one needs to use properties of linearity of expectation and scaling of variance. Calculating a z-score involves finding how many standard deviations a given value is from the mean of its distribution.
Step-by-step explanation:
The question posed involves understanding random variables and the relationship between the values these variables can take and their respective distributions. When dealing with random variables X, Y, and Z, we come across expected values, variance, and the concept of z-scores. In particular, the question seems to be asking how to compute the expected value and variance of a transformed random variable, and how to determine a z-score for a given value of another random variable in the context of normal distributions.
For example, if Y has an expected value (mean) E(Y) = 9 and someone is asking to compute E(-4Y+3), the linearity of expectation tells us to directly multiply the mean of Y by -4 and add 3. Similarly, for variance calculations, we should apply the property that if a random variable has a variance Var(Y), then Var(aY) = a2Var(Y), where 'a' is a constant.
As for z-scores, these are a measure of how many standard deviations an element is away from the mean. If we know that y = 4 corresponds to a z-score of 2 in its distribution, this simply means that the value of 4 is 2 standard deviations above the mean of that distribution. To find the z-score of any given value, we use the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation of the distribution.