To find the expected value of a random variable, you need to use the probabilities and corresponding scores provided and apply the formula for the expected value. The expected value, [tex]\(E(X)\)[/tex], is calculated by summing the products of each score and its corresponding probability. In mathematical terms:
[tex]\[ E(X) = \sum (P_i \times X_i) \][/tex]
where [tex]\(P_i\)[/tex] is the probability of score [tex]\(X_i\)[/tex].
Given the table:
[tex]\[
\begin{tabular}{|r|r|}
\hline Probability & Scores \\
\hline 0.16 & 1 \\
\hline 0.1 & 2 \\
\hline 0.18 & 4 \\
\hline 0.24 & 10 \\
\hline 0.19 & 13 \\
\hline 0.13 & 14 \\
\hline
\end{tabular}
\][/tex]
let’s find the expected value step by step:
1. Multiply each score by its probability:
[tex]\[
0.16 \times 1 = 0.16
\][/tex]
[tex]\[
0.1 \times 2 = 0.2
\][/tex]
[tex]\[
0.18 \times 4 = 0.72
\][/tex]
[tex]\[
0.24 \times 10 = 2.4
\][/tex]
[tex]\[
0.19 \times 13 = 2.47
\][/tex]
[tex]\[
0.13 \times 14 = 1.82
\][/tex]
2. Sum these products to get the expected value:
[tex]\[
0.16 + 0.2 + 0.72 + 2.4 + 2.47 + 1.82 = 7.77
\][/tex]
So, the expected value of the random variable is:
[tex]\[ E(X) = 7.77 \][/tex]
This value represents the average score you would expect if you were to repeat this random process many times.
