Given the following probability distribution:

[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 of this probability distribution is:
A. 1.5
B. 0.9
C. 0



Answer :

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]