To find the expected value (mean) of a random variable given its probability distribution, you use the formula:
[tex]\[
E(X) = \sum_{i} x_i \cdot P(x_i)
\][/tex]
where \( x_i \) are the values of the random variable and \( P(x_i) \) are the corresponding probabilities. Let's go through this step-by-step using the given values.
1. List the values of \( x \) and their probabilities \( P \):
[tex]\[
x = [2, 4, 6, 8, 10, 12]
\][/tex]
[tex]\[
P = [0.10, 0.34, 0.26, 0.16, 0.09, 0.05]
\][/tex]
2. Multiply each value of \( x \) by its corresponding probability:
[tex]\[
2 \times 0.10 = 0.20
\][/tex]
[tex]\[
4 \times 0.34 = 1.36
\][/tex]
[tex]\[
6 \times 0.26 = 1.56
\][/tex]
[tex]\[
8 \times 0.16 = 1.28
\][/tex]
[tex]\[
10 \times 0.09 = 0.90
\][/tex]
[tex]\[
12 \times 0.05 = 0.60
\][/tex]
3. Sum up all the products:
[tex]\[
0.20 + 1.36 + 1.56 + 1.28 + 0.90 + 0.60
\][/tex]
Adding these together:
[tex]\[
0.20 + 1.36 = 1.56
\][/tex]
[tex]\[
1.56 + 1.56 = 3.12
\][/tex]
[tex]\[
3.12 + 1.28 = 4.40
\][/tex]
[tex]\[
4.40 + 0.90 = 5.30
\][/tex]
[tex]\[
5.30 + 0.60 = 5.90
\][/tex]
Thus, the expected value \( \mu \) of the random variable is:
[tex]\[
\mu = 5.9
\][/tex]
