Mean Value of Random Variable

The mean value of a random variable is a measure of its central tendency. It represents the average value that the random variable takes on over a large number of trials.

In this question, we are given a random variable W which is defined as follows:

W = (X + 3Y)^2 - 2X + 3

We need to find the mean value of this random variable.

Breaking down the Random Variable

To find the mean value of a random variable, we need to calculate the expected value of that variable. In this case, we need to calculate the expected value of W.

To do this, we need to break down the random variable W into its constituent random variables X and Y.

W = (X + 3Y)^2 - 2X + 3

Expanding the square term, we get:

W = (X^2 + 6XY + 9Y^2) - 2X + 3

Simplifying further, we get:

W = X^2 + 6XY + 9Y^2 - 2X + 3

Now, we can calculate the expected value of W by taking the expected value of each term separately.

Calculating the Expected Value

To calculate the expected value, we need to know the probability distribution of the random variables X and Y. Without this information, we cannot calculate the mean value of W accurately.

However, we can make some assumptions about the probability distribution to simplify the calculation. Let's assume that X and Y are independent and identically distributed random variables with a mean of 0 and a variance of 1.

Using these assumptions, we can calculate the expected value of each term in the random variable W.

E(X^2) = Var(X) + E(X)^2 = 1 + 0^2 = 1
E(6XY) = E(X) * E(Y) * 6 = 0 * 0 * 6 = 0
E(9Y^2) = Var(Y) + E(Y)^2 = 1 + 0^2 = 1
E(-2X) = -2 * E(X) = -2 * 0 = 0
E(3) = 3

Summing up these expected values, we get:

E(W) = E(X^2) + E(6XY) + E(9Y^2) + E(-2X) + E(3)
= 1 + 0 + 1 + 0 + 3
= 5

Therefore, the mean value of the random variable W is 5.

Conclusion

The mean value of the random variable W, given by W = (X + 3Y)^2 - 2X + 3, is 5. Therefore, none of the given options (a), b), c), d)) is the correct answer.