\begin{tabular}{|c|r|} \hline$x$ & $P(x)$ \\ \hline 0 & 0.05 \\ \hline 1 & 0.15 \\ \hline 2 & 0.3 \\ \hline 3 & 0.5 \\ \hline \end{tabular} Find the mean of this probability distribution. Round your answer to one decimal place.

Step 1 ：The given probability distribution is as follows: \[\begin{tabular}{|c|r|} \hline$x$ & $P(x)$ \\ \hline 0 & 0.05 \\ \hline 1 & 0.15 \\ \hline 2 & 0.3 \\ \hline 3 & 0.5 \\ \hline \end{tabular}\]

Step 2 ：The mean of a probability distribution is calculated by multiplying each outcome by its probability and then summing these products. The formula for the mean is: \[\mu = \sum_{i=0}^{3} x_i \cdot P(x_i)\] where $x_i$ is the outcome and $P(x_i)$ is the probability of that outcome.

Step 3 ：Substitute the given values into the formula: \[\mu = (0 \times 0.05) + (1 \times 0.15) + (2 \times 0.3) + (3 \times 0.5)\]

Step 4 ：Simplify the expression to find the mean: \[\mu = 2.25\]

Step 5 ：Round the mean to one decimal place: \[\mu = 2.3\]

Step 6 ：Final Answer: The mean of this probability distribution is \(\boxed{2.3}\)