To find the mean of the given probability distribution, you use the formula for the expected value $E(X)$:

$E(X) = \sum (x_i \cdot P(x_i))$

Where:

• $x_i$ are the values of the random variable.
• $P(x_i)$ are their corresponding probabilities.

From the table in the image:

• $x_0 = 0$, $P(0) = 0.25$
• $x_1 = 1$, $P(1) = 0.15$
• $x_2 = 2$, $P(2) = 0.1$
• $x_3 = 3$, $P(3) = 0.5$

Now, we calculate the expected value $E(X)$.

$E(X) = (0 \cdot 0.25) + (1 \cdot 0.15) + (2 \cdot 0.1) + (3 \cdot 0.5)$

Let me compute that.The mean (expected value) of this probability distribution is $1.85$. When rounded to one decimal place, the mean is $1.9$.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the variance of this probability distribution?
2. How do you calculate the standard deviation from a probability distribution?
3. What does the mean represent in the context of probability distributions?
4. How can we verify that this is a valid probability distribution?
5. What is the probability that $X \geq 2$ based on this distribution?

Tip: In probability distributions, the mean represents the long-run average outcome after many trials.