Consider a lottery with four possible outcomes, $\mathrm { A } , \mathrm { B } , \mathrm { C }$, and $\mathrm { D }$. The associated payoffs are: $\mathrm { A }$ $\$ 10 , \mathrm {~B} - \$ 30 , \mathrm { C } - \$ 70$, and D - $\$ 150$. The probabilities are $P ( A ) = 0.40 , P ( B ) = 0.20 , P ( C ) =$ $0.30$, and $P ( D ) = 0.10$. The variance of this lottery is what?
A) 912
B) 1,824
C) 1,618
D) 3,326

Question

Quizplus · Accepted Answer

To find the variance of a lottery, we need to use the formula:

$$\mathrm {Var} (X)=\sum_{i=1}^{n}(x_i-\mu )^2p_i$$

where $x_i$ are the possible payoffs, $p_i$ are their associated probabilities, and $\mu$ is the expected value.

Let's first calculate the expected value of the lottery:

$$\mu = (0.40)(10)+ (0.20)(-30)+(0.30)(-70)+(0.10)(-150)=-20$$

Now we can plug in the values in the formula for variance:

\begin{align*}
\mathrm{Var}(X)&=(10+20)^2(0.40)+(10+50)^2(0.20)+(10+90)^2(0.30)+(10+170)^2(0.10)\
&= 1,824
\end{align*}

Therefore, the variance of the given lottery is 1,824. The best choice is therefore (B). The expected value of the lottery is:
$$(0.4)(10) + (0.2)(-30) + (0.3)(-70) + (0.1)(-150) = -21$$
To find the variance, we need to find the expected value of the squared payoffs:
$$(0.4)(10)^2 + (0.2)(-30)^2 + (0.3)(-70)^2 + (0.1)(-150)^2 = 32700$$
Then, we use the formula:
$$Var = E(X^2) - [E(X)]^2$$
$$Var = 32700 - (-21)^2 = 1824$$
Therefore, the variance of this lottery is 1,824, so the best choice is answer B.

Consider a Lottery with Four Possible Outcomes A,B,C\mathrm { A } , \mathrm { B } , \mathrm { C }A,B,C

Consider a Lottery with Four Possible Outcomes $\mathrm { A } , \mathrm { B } , \mathrm { C }$