Find the expected mean of this probability distribution.
| x | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| p(x) | 1 / 5 | 1 / 5 | 1 / 5 | 1 / 5 | 1 / 5 |
A. 5.6
B. 6
C. 4
D. 3.8
Answer (2)
The expected mean is calculated by summing the product of each value and its probability.
The formula for expected mean is: E ( X ) = ∑ i = 1 n x i ∗ p ( x i ) .
Applying the formula to the given data: E ( X ) = ( 0 ∗ 5 1 ) + ( 2 ∗ 5 1 ) + ( 4 ∗ 5 1 ) + ( 6 ∗ 5 1 ) + ( 8 ∗ 5 1 ) .
The expected mean is: 4 .
Explanation
Understand the problem and provided data We are given a discrete probability distribution with values x = { 0 , 2 , 4 , 6 , 8 } and corresponding probabilities p ( x ) = { 5 1 , 5 1 , 5 1 , 5 1 , 5 1 } . The goal is to find the expected mean of this distribution.
State the formula for expected mean The expected mean (or expected value) of a discrete probability distribution is calculated as the sum of each value multiplied by its probability. The formula is: E ( X ) = i = 1 ∑ n x i ∗ p ( x i ) where x i are the values and p ( x i ) are their corresponding probabilities.
Apply the formula to the given data In this case, we have: E ( X ) = ( 0 ∗ 5 1 ) + ( 2 ∗ 5 1 ) + ( 4 ∗ 5 1 ) + ( 6 ∗ 5 1 ) + ( 8 ∗ 5 1 )
Calculate the expected mean Now, let's simplify the expression: E ( X ) = 0 + 5 2 + 5 4 + 5 6 + 5 8 E ( X ) = 5 2 + 4 + 6 + 8 E ( X ) = 5 20 E ( X ) = 4
State the final answer The expected mean of the given probability distribution is 4.
Examples
The expected value is a fundamental concept in finance, used to estimate the potential return of an investment. For example, if you're investing in a stock, you can create a probability distribution of potential returns (e.g., 10% gain with 0.3 probability, 5% loss with 0.2 probability, etc.). The expected value then gives you a single number representing the average return you can expect from the investment, helping you compare different investment opportunities. The calculation is performed by multiplying each possible outcome by its probability and summing the results: E ( X ) = ∑ x i p i .
The expected mean of the given probability distribution is calculated to be 4 using the formula for expected value. By summing the products of each value and its probability, we arrive at the final result. Therefore, the correct choice is C. 4.
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