Let's calculate the expected value of the given probability distribution step-by-step.
The table provided lists the values of [tex]\( x \)[/tex] and their corresponding probabilities [tex]\( P(x) \)[/tex]:
[tex]\[
\begin{tabular}{c|cccc}
x & 0 & 1 & 2 & 3 \\
\hline
P(x) & 0.5 & 0.2 & 0.2 & 0.1 \\
\end{tabular}
\][/tex]
The expected value [tex]\( E(X) \)[/tex] of a discrete random variable is calculated using the formula:
[tex]\[
E(X) = \sum_{i} x_i \cdot P(x_i)
\][/tex]
Where [tex]\( x_i \)[/tex] are the values of the random variable and [tex]\( P(x_i) \)[/tex] are their corresponding probabilities.
Let's break down the calculation:
1. Multiply each value of [tex]\( x \)[/tex] by its corresponding probability [tex]\( P(x) \)[/tex]:
[tex]\[
0 \cdot 0.5 = 0
\][/tex]
[tex]\[
1 \cdot 0.2 = 0.2
\][/tex]
[tex]\[
2 \cdot 0.2 = 0.4
\][/tex]
[tex]\[
3 \cdot 0.1 = 0.3
\][/tex]
2. Sum these products to get the expected value:
[tex]\[
E(X) = 0 + 0.2 + 0.4 + 0.3
\][/tex]
3. Thus, the expected value calculation leads to:
[tex]\[
E(X) = 0.9
\][/tex]
Therefore, the expected value of this probability distribution is:
[tex]\[
\boxed{0.9}
\][/tex]
