Question

Let (X) be a random variable that takes nonnegative integer values, and is associated with a transform of the form (Y = cX), where (c) is some scalar. Find:

a) (EX)
b) (P_X(1))
c) (EX ⋅ Y)

Answer 1

A random variable X represents numerical outcomes of a random phenomenon, and the given problem involves finding its expected value, probability of taking the value 1, and expected product with its transformation. Without the distribution of X, we cannot calculate these values, but definitions and formulas would typically be used to find them given the distribution.

Step-by-step explanation:

To answer the student's question concerning a random variable X and its transformation Y = cX, we need to clarify the following:

1. In words, a random variable X is a variable whose possible values are numerical outcomes of a random phenomenon.
2. The values that X may take on are typically a set of nonnegative integers if X represents something countable like the number of occurrences.
3. The distribution of X has not been explicitly given in the question, so we cannot provide an exact probability distribution without additional information.
4. The expectation of X (EX) is the average or mean value of X, which is calculated as the sum of all possible values of X multiplied by their respective probabilities. Without the distribution of X, we cannot calculate EX.
5. The probability that X equals 1 (P_X(1)) is the probability that the random variable X will take the value 1. Again, without the distribution of X, this probability cannot be calculated.
6. The expectation of the product EX⋅Y (EX⋅Y) is the expected value of the product of X and Y. If Y = cX, then this expectation is the expectation of X multiplied by c, or c(EX)^2, assuming independence between X and Y. Without knowing the distribution or specific value of c, we cannot provide a numerical answer.

Given that the distribution of X is not provided, it would be speculation to offer numerical answers to parts a), b), and c) of the original question. However, if a student needs to be guided through finding these values, given the distribution, one would need to use the definitions and formulas pertaining to expectation and probability for discrete random variables.