Assuming that the returns from holding small-company stocks are normally distributed, what is the approximate probability that your money will double in value in a single year? What about triple in value.

Many of my classmates have an expected return of 17.6% and a standard deviation of 34.8. Then they have the formula Prob(d>(200-17.6)/34.8 = prob. d>2.37
1-d>2.37=1-.991 = .009 chance of doubling.

I just need this problem explained to me. I'm not sure where all the numbers and information are coming from. Any help would be greatly appreciated.

normal distribution problem

I do not know where your mean (expected return) and sigma are coming from. I assume some text book.

Now if we assume a mean return of 17.6%

and a sigma of 34.8%
doubling is 100% which is 82.4% above mean and tripling is 200% which is 182.4% above mean
how many sigmas is 82.4?
82.4/34.8 =2.37 sigmas to double
how many sigmas is 182.4?
182.4/34.8 = 5.24 sigmas to triple
(note you used 200 where you should have used 100 in interpreting classmate results)

Now go to tables for normal distribution:

to be 2.37 sigmas above mean:
well I only have a rough table here. For z = 2.3, F(z) = .989
for z = 2.4, F(z) = .992
so about 99% is below 2.37 sigma above mean and we only have about a 100-99 or a 1 percent chance of doubling. (you probably have more accurate tables). You will need a very accurate table though to find any probability of exceeding 5.24 sigmas above mean

To understand the problem, we'll break it down step by step.

1. The problem assumes that the returns from holding small-company stocks follow a normal distribution. In finance, a normal distribution is a statistical concept that represents the probability distribution of a random variable. It is often used to model and analyze uncertain outcomes.

2. The problem asks for the probability that your money will double in value in a single year. This means that you want to know the likelihood of getting a return of 100% (i.e., doubling your initial investment) from holding small-company stocks.

3. The formula used by your classmates is: Prob(d > (200 - 17.6) / 34.8) where d represents the standard deviation of the distribution. The numerator, (200 - 17.6), represents the difference between the desired return (200% or doubling the initial investment) and the expected return (17.6%). The denominator, 34.8, represents the standard deviation of the returns.

4. The calculation (200 - 17.6) / 34.8 gives you the z-score, which is a measure of how many standard deviations a particular value is from the mean. In this case, it tells you how many standard deviations away a return of 200% is from the mean return of 17.6%.

5. The next step is to find the probability of obtaining a value greater than the z-score. This is done by looking up the z-score in a standard normal distribution table or by using software or calculators that can perform z-score calculations. In your classmates' case, they found that the probability was 0.009 or 0.9%.

6. Finally, to find the probability of tripling your money, you would follow a similar process. You would use the formula Prob(d > (300 - 17.6) / 34.8) to calculate the z-score, and then find the probability of obtaining a value greater than the z-score.

It's important to note that this calculation assumes a normal distribution for the returns of small-company stocks, which might not always be the case in reality. Additionally, these calculations are based on the assumptions of the given expected return and standard deviation.