Questions: The following table provides a probability distribution for the random variable y. y f(y) ------ 2 0.10 4 0.40 7 0.20 8 0.30 (a) Compute E(y). E(y)= (b) Compute Var(y) and σ. (Round your answer for σ to two decimal places.) Var(y) = σ =

The following table provides a probability distribution for the random variable y.

y f(y)
------
2 0.10
4 0.40
7 0.20
8 0.30

(a) Compute E(y).

E(y)=

(b) Compute Var(y) and σ. (Round your answer for σ to two decimal places.)

Var(y) =

σ =

Transcript text: The following table provides a probability distribution for the random variable $y$. \begin{tabular}{|l|l|} \hline $\boldsymbol{y}$ & $\boldsymbol{f}(\boldsymbol{y})$ \\ \hline 2 & 0.10 \\ \hline 4 & 0.40 \\ \hline 7 & 0.20 \\ \hline 8 & 0.30 \\ \hline \end{tabular} (a) Compute $E(y)$. \[ E(y)= \] (b) Compute $\operatorname{Var}(y)$ and $\sigma$. (Round your answer for $\sigma$ to two decimal places.) \[ \begin{aligned} \operatorname{Var}(y) & =\square \\ \sigma & =\square \end{aligned} \]

Solution

Solution Steps

Step 1: Calculate the Mean $ E(y) $

The expected value $ E(y) $ is calculated using the formula:

\[ E(y) = \sum (y \cdot f(y)) = 2 \times 0.1 + 4 \times 0.4 + 7 \times 0.2 + 8 \times 0.3 \]

Calculating each term:

\[ E(y) = 0.2 + 1.6 + 1.4 + 2.4 = 5.6 \]

Step 2: Calculate the Variance $ \operatorname{Var}(y) $

The variance $ \operatorname{Var}(y) $ is calculated using the formula:

\[ \operatorname{Var}(y) = \sum ((y - E(y))^2 \cdot f(y)) \]

Substituting $ E(y) = 5.6 $:

\[ \operatorname{Var}(y) = (2 - 5.6)^2 \times 0.1 + (4 - 5.6)^2 \times 0.4 + (7 - 5.6)^2 \times 0.2 + (8 - 5.6)^2 \times 0.3 \]

Calculating each term:

\[ = (3.6)^2 \times 0.1 + (1.6)^2 \times 0.4 + (1.4)^2 \times 0.2 + (2.4)^2 \times 0.3 \] \[ = 12.96 \times 0.1 + 2.56 \times 0.4 + 1.96 \times 0.2 + 5.76 \times 0.3 \] \[ = 1.296 + 1.024 + 0.392 + 1.728 = 4.44 \]

Step 3: Calculate the Standard Deviation $ \sigma $

The standard deviation $ \sigma $ is the square root of the variance:

\[ \sigma = \sqrt{\operatorname{Var}(y)} = \sqrt{4.44} \approx 2.11 \]