The question asks for the expected value of x given a discrete probability distribution.

Calculating Expected Value E(x)

The expected value, denoted as E(x), for a discrete random variable is calculated using the formula:

$ E(x) = \sum_{i} x_i P(x_i) $

Where x_i represents each possible value of the variable x, and P(x_i) is the probability of that value occurring.

Applying the Formula to the Dataset

From the given dataset:

We can calculate the expected value:

$ E(x) = (2 \times \frac{1}{4}) + (3 \times \frac{1}{2}) + (4 \times \frac{1}{4}) $

Step-by-Step Calculation

1. Multiply each x value by its corresponding probability f(x):
   • $2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$
   • $3 \times \frac{1}{2} = \frac{3}{2}$
   • $4 \times \frac{1}{4} = \frac{4}{4} = 1$
2. Sum the results from step 1:

   $ E(x) = \frac{1}{2} + \frac{3}{2} + 1 $

3. Simplify the sum:

   $ E(x) = \frac{1+3}{2} + 1 = \frac{4}{2} + 1 = 2 + 1 = 3 $

Conclusion

The calculated expected value of x is 3.