To find the expected value [tex]\(\mu\)[/tex] of the random variable given the probability distribution, follow these steps:
1. Understand the definition of expected value: The expected value (or mean) of a discrete random variable is the sum of the products of each value of the random variable and its corresponding probability.
2. Set up the formula: If [tex]\(X\)[/tex] is a random variable with values [tex]\(x\)[/tex] and corresponding probabilities [tex]\(P(x)\)[/tex], the expected value [tex]\(\mu\)[/tex] is calculated as:
[tex]\[
\mu = \sum_{i} x_i P(x_i)
\][/tex]
where [tex]\(x_i\)[/tex] are the values of the random variable, and [tex]\(P(x_i)\)[/tex] are the corresponding probabilities.
3. List the values and probabilities: From the given table, we have:
[tex]\[
\begin{array}{c|c}
x & P(x) \\
\hline
1 & 0.11 \\
3 & 0.19 \\
5 & 0.26 \\
7 & 0.21 \\
9 & 0.14 \\
11 & 0.09 \\
\end{array}
\][/tex]
4. Multiply each value by its probability and sum the products:
[tex]\[
\begin{align*}
1 \times 0.11 &= 0.11, \\
3 \times 0.19 &= 0.57, \\
5 \times 0.26 &= 1.30, \\
7 \times 0.21 &= 1.47, \\
9 \times 0.14 &= 1.26, \\
11 \times 0.09 &= 0.99.
\end{align*}
\][/tex]
5. Sum up all the products:
[tex]\[
0.11 + 0.57 + 1.30 + 1.47 + 1.26 + 0.99 = 5.7
\][/tex]
Therefore, the expected value [tex]\(\mu\)[/tex] of the random variable is:
[tex]\[
\mu = 5.7
\][/tex]
