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)E[X]=∑xi​×P(xi​)
Where xix_ixi​
equal to the values that XXX
takes and P(xi)P(x_i)P(xi​)
 is the probability that XXX
 takes the value xix_ixi​
 
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) dxE[X]=∫xmin​xmax​​x×f(x)dx
Where f(x)f(x)f(x)
 is the PDF of XXX
​

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Last modified 2yr ago