Suppose that U is a random number with mean 10, and V is a random number with mean 6. What is the expected value of U+V?
6
10
16
Not enough information given
The correct answer and explanation is :
The correct answer is 16.
Explanation:
In probability theory and statistics, the expected value of a random variable is essentially its mean or average value. If we have two random variables, $U$ and $V$, the expected value of their sum can be determined using a fundamental property of expected values.
Property of Expected Value:
One important property of expected values is that the expected value of the sum of two random variables is the sum of their expected values. This is written mathematically as:
$$
E(U + V) = E(U) + E(V)
$$
Where:
- $E(U)$ represents the expected value (or mean) of the random variable $U$,
- $E(V)$ represents the expected value (or mean) of the random variable $V$,
- $E(U + V)$ represents the expected value of the sum of $U$ and $V$.
Given Information:
- The mean (or expected value) of $U$ is 10, so $E(U) = 10$,
- The mean (or expected value) of $V$ is 6, so $E(V) = 6$.
Applying the Property:
Using the property mentioned above, the expected value of the sum $U + V$ is:
$$
E(U + V) = E(U) + E(V)
$$
$$
E(U + V) = 10 + 6 = 16
$$
So, the expected value of $U + V$ is 16.
Key Points:
- The expected value of a sum of two independent (or dependent) random variables is simply the sum of their expected values.
- In this case, $U$ and $V$ could be either independent or dependent, but regardless of their relationship, the expected value of $U + V$ is 16 based on the given means.
- The formula relies on the fact that expected value is a linear operator, meaning it respects the addition of random variables.
This property is widely used in statistics and probability to calculate the expected value of sums of random variables, and it holds regardless of whether the variables are independent or not.