To find the mean of the given probability distribution, we follow these steps:
1. Identify the random variable values and their corresponding probabilities:
[tex]\[
\begin{array}{|r|r|}
\hline
x & P(x) \\
\hline
0 & 0.25 \\
\hline
1 & 0.3 \\
\hline
2 & 0.3 \\
\hline
3 & 0.15 \\
\hline
\end{array}
\][/tex]
2. Calculate the mean ([tex]\(\mu\)[/tex]) of the probability distribution:
The formula for the mean of a discrete probability distribution is given by:
[tex]\[
\mu = \sum (x_i \cdot P(x_i))
\][/tex]
Where [tex]\( x_i \)[/tex] are the possible values of the random variable [tex]\( x \)[/tex], and [tex]\( P(x_i) \)[/tex] are their corresponding probabilities.
3. Compute the products [tex]\( x_i \cdot P(x_i) \)[/tex] for each [tex]\( x_i \)[/tex] and [tex]\( P(x_i) \)[/tex]:
[tex]\[
0 \cdot 0.25 = 0
\][/tex]
[tex]\[
1 \cdot 0.3 = 0.3
\][/tex]
[tex]\[
2 \cdot 0.3 = 0.6
\][/tex]
[tex]\[
3 \cdot 0.15 = 0.45
\][/tex]
4. Sum these products to find the mean:
[tex]\[
\mu = 0 + 0.3 + 0.6 + 0.45 = 1.3499999999999999
\][/tex]
5. Round the mean to one decimal place:
[tex]\[
\mu \approx 1.3
\][/tex]
So, the mean of the given probability distribution is approximately 1.3 when rounded to one decimal place.
