To find the expected value $E(X)$, we use the formula for the expected value of a discrete random variable:

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

Where:

• $x$ are the values of the random variable,
• $P(X = x)$ is the probability associated with each value of $x$.

From the given data:

• $x = 3, 4, 5, 6, 7$
• $P(X = x) = 0.1, 0.3, 0.3, 0.1, 0.2$

Now, calculate $E(X)$:

$E(X) = (3 \cdot 0.1) + (4 \cdot 0.3) + (5 \cdot 0.3) + (6 \cdot 0.1) + (7 \cdot 0.2)$ $E(X) = 0.3 + 1.2 + 1.5 + 0.6 + 1.4$ $E(X) = 5.0$

So, the expected value $E(X)$ is 5.0.

Let me know if you need more details or if you have any questions!

Related questions:

1. What is the variance of the distribution for this data?
2. How would the expected value change if the probability distribution was different?
3. Can the expected value ever be outside the range of the given values of $x$?
4. What is the standard deviation of this data set?
5. How would you calculate the expected value if the probabilities were given as percentages instead of decimals?

Tip: The expected value gives a "weighted average" of all possible values, accounting for how likely each value is to occur.