Answered

Using the data from the table, answer the following questions:

\begin{tabular}{|c|c|}
\hline
[tex]$X$[/tex] & Probability: [tex]$P(X)$[/tex] \\
\hline
0 & 0.1 \\
\hline
1 & 0.2 \\
\hline
2 & 0.4 \\
\hline
3 & 0.2 \\
\hline
4 & 0.1 \\
\hline
\end{tabular}

1. What is [tex]$P(3)$[/tex]?

A. 0.2

2. What is the mean of the probability distribution?

[tex]$\square$[/tex]



Answer :

GinnyAnswer
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]