O retorno mensal de certo investimento de risco pode ser modelado pela variável aleatória W, com função de probabilidade dada a seguir. mathrm(W) & -5 % & 0 % & 5 % & 10 % & 15 % mathrm(P)(mathrm(W)=mathrm(w)) & 0,4 & 0,15 & 0,25 & 0,15 & 0,05 O retorno esperado é: A -0,5 % . B 0,5 % . C 1,5 % . D 5 % . E 7,5 % .

## Step 1: Understand the Problem<br />### We are given a probability distribution for the monthly return \( W \) of a risky investment. The table provides possible returns and their associated probabilities. We need to calculate the expected return.<br /><br />## Step 2: Define the Expected Value Formula<br />### The expected value \( E(W) \) of a discrete random variable is calculated using the formula:<br />\[<br />E(W) = \sum_{i} w_i \cdot P(W = w_i)<br />\]<br />where \( w_i \) are the possible values of \( W \), and \( P(W = w_i) \) are their respective probabilities.<br /><br />## Step 3: Apply the Values from the Table<br />### Using the values from the table:<br />- \( w_1 = -5\% \), \( P(W = -5\%) = 0.4 \)<br />- \( w_2 = 0\% \), \( P(W = 0\%) = 0.15 \)<br />- \( w_3 = 5\% \), \( P(W = 5\%) = 0.25 \)<br />- \( w_4 = 10\% \), \( P(W = 10\%) = 0.15 \)<br />- \( w_5 = 15\% \), \( P(W = 15\%) = 0.05 \)<br /><br />## Step 4: Calculate the Expected Return<br />### Substitute these values into the expected value formula:<br />\[<br />E(W) = (-5\% \times 0.4) + (0\% \times 0.15) + (5\% \times 0.25) + (10\% \times 0.15) + (15\% \times 0.05)<br />\]<br /><br />### Simplify each term:<br />- \(-5\% \times 0.4 = -2\%\)<br />- \(0\% \times 0.15 = 0\%\)<br />- \(5\% \times 0.25 = 1.25\%\)<br />- \(10\% \times 0.15 = 1.5\%\)<br />- \(15\% \times 0.05 = 0.75\%\)<br /><br />### Sum these results:<br />\[<br />E(W) = -2\% + 0\% + 1.25\% + 1.5\% + 0.75\% = 1.5\%<br />\]