To find the mean of the probability distribution and determine [tex]\( P(3) \)[/tex] given the following data:
[tex]\[
\begin{tabular}{|c|c|}
\hline
X & Probability: \( P(X) \) \\
\hline
0 & 0.1 \\
\hline
1 & 0.2 \\
\hline
2 & 0.4 \\
\hline
3 & 0.2 \\
\hline
4 & 0.1 \\
\hline
\end{tabular}
\][/tex]
Let's break this down step by step.
Step 1: Finding [tex]\( P(3) \)[/tex]
From the table, we see that the probability associated with [tex]\( X = 3 \)[/tex] is:
[tex]\[
P(3) = 0.2
\][/tex]
Step 2: Calculating the Mean (Expected Value)
The mean [tex]\( \mu \)[/tex] of a discrete probability distribution is given by the formula:
[tex]\[
\mu = \sum_{i} (X_i \cdot P(X_i))
\][/tex]
Using the data provided:
[tex]\[
\begin{align*}
\mu &= (0 \times 0.1) + (1 \times 0.2) + (2 \times 0.4) + (3 \times 0.2) + (4 \times 0.1) \\
&= 0 + 0.2 + 0.8 + 0.6 + 0.4 \\
&= 2.0
\end{align*}
\][/tex]
The mean of the probability distribution is therefore:
[tex]\[
\mu = 2.0
\][/tex]
In summary:
- [tex]\( P(3) = 0.2 \)[/tex]
- The mean of the probability distribution is [tex]\( 2.0 \)[/tex]
