a) Find the expected value of 8 minus the random variable X.
b) Find the expected value of the product (X minus 3) multiplied by (X plus 3).
To find the expected value of a random variable, you need to know its probability distribution. Only then can you calculate the expected value by taking the sum of the product of each possible value of the random variable and its corresponding probability.
a) To find the expected value of 8 minus the random variable X, you would need to know the probability distribution of X. Let's assume X follows a discrete probability distribution. If you have the probability mass function (PMF) or probability distribution table for X, you would need to calculate:
E[8 - X] = ∑[x(P(X=x))] where the summation is taken over all possible values of x.
If X is continuous, you would need the probability density function (PDF), and the integral for the expected value is calculated in a similar manner:
E[8 - X] = ∫[x(f(x))]dx where the integral is taken over the entire range of x.
b) The expected value of the product (X - 3) multiplied by (X + 3) can be found using the same principles. You would need the probability distribution of X to calculate it. The formula would be:
E[(X - 3)(X + 3)] = ∑[(x - 3)(x + 3)(P(X=x))] or ∫[(x - 3)(x + 3)(f(x))]dx, depending on whether X is discrete or continuous.
Remember that finding the expected value requires knowledge of the probability distribution of the random variable.