Exercise 17 Page 748

To advise Carmen which investment to choose, we will first find the expected values and the standard deviations of both investments. Let's start with calculating the expected value of Fund A.

Fund A

The expected value of a discrete random variable X is a weighted sum of its values. E(X)=∑ [ X * P(X)] In our case, the values of X will represent the profit of Fund A. We will think of losses as negative profits. Let's look at the given information about Fund A.

To calculate the expected value, we need to add up all the products of the profits and their corresponding probabilities. Before we start calculating, let's convert the probabilities from percent to decimal notation for convenience.

Now, we can calculate all the products of the profits and their probabilities.

To find the expected value of Fund A we need to sum all values from the third column. E(X)=$570 + $180 +(-$30) +(-$125) =$595The expected value of Fund A is $595. Next, let's find the the standard deviation of Fund A. The standard deviation σ of a random variable X is given by the following formula. σ=sqrt(∑ [[X-E(X)]^2* P(X)]) To calculate the standard deviation, we will first calculate the expression inside the radical, which is called the variance of a random variable X. σ^2=∑ [[X-E(X)]^2* P(X)] Earlier, we found that E(X)=$ 595. Let's substitute that value into the equation. σ^2=∑ [[X- 595]^2* P(X)] The variance σ^2 is a sum of elements. Let's calculate each element of the sum one at the time.

Now, we can find the variance by summing all the elements from the last column. Let's do it! σ^2=510 907.5+7.5+94 803.75+ +299 756.25= 905 475 Finally, we can calculate the standard deviation by calculating the square root of the variance σ^2.

We found that the standard deviation of Fund A is 951.6.

Fund B

Next, we can calculate the expected value of Fund B. We will follow the same process as in the previous part. First we will look at the given information.

Now, we can calculate all the products of profits and their probabilities.

The expected value of Fund B is the sum of all values from the last column. E(X)=$640+$90+(-$30)+(-$160) =$540 The expected value of Fund B is $540. Now, lets calculate the standard deviation σ of Fund B. As before, we will first calculate the variance σ^2. σ^2=∑ [[X-E(X)]^2* P(X)] We can substitute E(X)=$ 540 into the equation. σ^2=∑ [[X- 540]^2* P(X)] Let's calculate each element of the sum.

To calculate the variance σ^2 we need to sum all elements from the last column. Here we go! σ^2=449 440+12 960+ +70 560+353 440=886 400 Finally, to find the standard deviation σ of Fund B we can calculate the square root of the variance σ^2.

The standard deviation of Fund B is 941.5. We found all the information we needed to compare the funds.

Comparison of the Funds

Let's summarize all the information we found.

The standard deviations are very similar, which means that both funds have the same amount of risk involved. Therefore, the better fund is the one with a higher expected value. Since Fund A has a higher expected value, it is the one we should advise Carmen to invest in.