Final answer:
The expected value E(Y) of a random integer Y selected from {1,2,...X}, with X itself randomly chosen from {1,2,...,n}, is calculated by averaging over all possible values of X to yield a final result of (n+1)/4, assuming a uniform distribution for X.
Step-by-step explanation:
The student's question revolves around finding the expected value E(Y) where Y is a random integer selected uniformly from the set {1,2,...X}, and X itself is a randomly selected label from the set {1,2,...,n}. To approach this, we have to acknowledge that the first action is choosing an integer X, which affects the range of possible values for Y. The expected value E(Y) can be understood as the long-term average of the values of Y if this experiment is repeated many times.
To find the expected value of Y, we need to consider that Y could be any integer between 1 and X, so its expected value given X is X/2 (since the average of a set of integers from 1 to X is X/2). However, since X can vary from 1 to n, we must average over all possible values of X to get the overall expected value of Y for this two-step process. Using the definition of expected value, E(Y) would be the sum of X/2 multiplied by the probability of each value of X.
The probability of selecting any particular value of X is 1/n since we are given that X is selected randomly from {1,2,...,n}. Therefore, E(Y) can be calculated by summing up all the expected values of Y given each X and then dividing by n. So, the final formula for E(Y) is the sum from x=1 to n of (x/2) times (1/n), which simplifies to (n+1)/4. This calculation assumes a uniform distribution over the initial set of X values.