\begin{tabular}{l|llll}
$P(x)$ & 0.2 & 0.4 & 0.3 & 0.1
\end{tabular}
What is the expected value?
Answer (1)
The problem provides a probability distribution P(x) = [0.2, 0.4, 0.3, 0.1] for x = [1, 2, 3, 4].
The expected value E[x] is calculated as the sum of each value multiplied by its probability.
Calculate the sum: E [ x ] = ( 1 ∗ 0.2 ) + ( 2 ∗ 0.4 ) + ( 3 ∗ 0.3 ) + ( 4 ∗ 0.1 ) = 0.2 + 0.8 + 0.9 + 0.4 .
The expected value is 2.3 .
Explanation
Understand the problem and provided data We are given a probability distribution P ( x ) with the following values:
P ( x ) = [ 0.2 , 0.4 , 0.3 , 0.1 ]
We need to find the expected value of this probability distribution. The values of x are implicitly assumed to be [ 1 , 2 , 3 , 4 ] since there are 4 probabilities given.
Define the expected value The expected value E [ x ] is calculated as the sum of each x i multiplied by its corresponding probability P ( x i ) .
E [ x ] = ∑ ( x i ∗ P ( x i ))
for i = 1 to 4 .
Calculate the expected value Now, we calculate the expected value:
E [ x ] = ( 1 ∗ 0.2 ) + ( 2 ∗ 0.4 ) + ( 3 ∗ 0.3 ) + ( 4 ∗ 0.1 )
E [ x ] = 0.2 + 0.8 + 0.9 + 0.4
E [ x ] = 2.3
State the final answer The expected value of the given probability distribution is 2.3 .
Examples
Expected value is a fundamental concept in finance and investment. For example, if you're considering investing in a stock, you can use the probabilities of different market conditions (boom, normal, recession) and the potential returns in each scenario to calculate the expected return of the stock. This helps you make informed investment decisions by quantifying the average outcome you can expect over the long run.
