<\/span><\/h2>\nThe expected value of a function, E(f(x)), is calculated by multiplying each possible value of the random variable x by its corresponding probability, and then summing these products. It can be mathematically represented as:<\/p>\n
E(f(x)) = \u03a3 f(x) * P(x)<\/p>\n
Where \u03a3 denotes the sum, f(x) represents the function applied to the random variable x, and P(x) is the probability of x occurring.<\/p>\n
To find the expected value of f(x), follow these steps:<\/b><\/p>\n
Step 1: Identify the function f(x) that you want to evaluate.<\/p>\n
Step 2: Determine the probability distribution of the random variable x. This can be given through a probability density function (PDF) or a probability mass function (PMF), depending on whether x is continuous or discrete, respectively.<\/p>\n
Step 3: Calculate f(x) for each possible value of x.<\/p>\n
Step 4: Multiply each value of f(x) by its corresponding probability. This can be done by multiplying f(x) by P(x) for discrete random variables or by integrating f(x) * P(x) over the range of x for continuous random variables.<\/p>\n
Step 5: Sum all the products calculated in the previous step. This sum represents the expected value of f(x).<\/p>\n
For example:<\/b><\/p>\n
Suppose you want to find the expected value of the function f(x) = x^2 for a discrete random variable x.<\/p>\n
1. Identify the function f(x) as x^2.
\n2. Determine the probability distribution of x, let’s say it is given by P(x) = {0.2, 0.4, 0.4}.
\n3. Calculate f(x) for each possible value of x: f(1) = 1^2 = 1, f(2) = 2^2 = 4, and f(3) = 3^2 = 9.
\n4. Multiply each value of f(x) by its corresponding probability: 0.2 * 1 + 0.4 * 4 + 0.4 * 9 = 0.2 + 1.6 + 3.6 = 5.4.
\n5. The expected value of f(x) is 5.4.<\/p>\n