Expected Value

Expected Value of Random Variable

The expected value of a random variable is the weighted average of all possible values of the variable. The weight here means the probability of the random variable taking a specific value.

Expected Value of Discrete Random Variable

E[X]=xi×P(xi)E[X] = \sum x_i \times P(x_i)

Where xix_iequal to the values that XXtakes and P(xi)P(x_i) is the probability that XX takes the value xix_i

Expected Value of Continuous Random Variable

Since continuous random variables can take uncountably infinitely many values, we cannot talk about a variable taking a specific value. We rather focus on value ranges.

In order to calculate the probability of value ranges, probability density functions (PDF) are used.

PDF is a function that specifies the probability of a random variable taking value within a particular range.

E[X]=xminxmaxx×f(x)dxE[X] = \int_{x_{min}}^{x_{max}} x \times f(x) dx

Where f(x)f(x) is the PDF of XX