Use the following two probability distributions for sales of a firm to answer the following question: $$\begin{array} { c c c }
\text { Sales } & \begin{array} { c }
\text { Disilibisiont } \
\text { Probabildy }
\end{array} & \begin{array} { c }
\text { Dilribulion2 } \
\text { Probabilly }
\end{array} \
\hline 2,000 & 0.05 & 0.05 \
3,000 & 0.20 & 0.15 \
4,000 & 0.50 & 0.20 \
5,000 & 0.20 & 0.35 \
6,000 & 0.05 & 0.25
\end{array}$$ Which distribution is more risky?
A)Distribution 1 has a higher variance than Distribution 2,so Distribution 1 is more risky.
B)Distribution 2 has a higher variance than Distribution 1,so Distribution 2 is more risky.
C)Distribution 2 has a larger standard deviation than Distribution 1,so Distribution 2 is more risky.
D)both b and c

Question

Use the following two probability distributions for sales of a firm to answer the following question: $$\begin{array} { c c c } 
\text { Sales } & \begin{array} { c } 
\text { Disilibisiont } \
\text { Probabildy }
\end{array} & \begin{array} { c } 
\text { Dilribulion2 } \
\text { Probabilly }
\end{array} \
\hline 2,000 & 0.05 & 0.05 \
3,000 & 0.20 & 0.15 \
4,000 & 0.50 & 0.20 \
5,000 & 0.20 & 0.35 \
6,000 & 0.05 & 0.25
\end{array}$$ Which distribution is more risky?
A)Distribution 1 has a higher variance than Distribution 2,so Distribution 1 is more risky.
B)Distribution 2 has a higher variance than Distribution 1,so Distribution 2 is more risky.
C)Distribution 2 has a larger standard deviation than Distribution 1,so Distribution 2 is more risky.
D)both b and c

Quizplus · Accepted Answer

To determine which distribution is more risky, we need to compare their variances or standard deviations. The variance of Distribution 1 is:
$$Var(X) = \sum (x_i - \mu)^2 P(X = x_i) = (2,000-3,800)^2(0.05) + \cdots +(6,000-3,800)^2(0.05) \approx 1,760,000$$
The variance of Distribution 2 is:
$$Var(Y) = \sum (y_i - \mu)^2 P(Y = y_i) = (2,000-4,100)^2(0.05) + \cdots +(6,000-4,100)^2(0.25) \approx 2,245,000$$
Therefore, Distribution 2 has a higher variance than Distribution 1, making it more risky. Additionally, the standard deviation of Distribution 1 is approximately $\sqrt{1,760,000} \approx 1,327$, while the standard deviation of Distribution 2 is approximately $\sqrt{2,245,000} \approx 1,498$. Thus, Distribution 2 also has a larger standard deviation.

Use the Following Two Probability Distributions for Sales of a Firm