Probability of a normally distributed random variable inside a given interval

CFA level I / Quantitative Methods: Application / Common Probability Distributions / Probability of a normally distributed random variable inside a given interval

The calculation of the probability of a normally distributed random variable inside a given interval is similar to that of uniform continuous distribution. The only difference is that in the normal distribution, the probabilities will not be same for all the outcomes. The similarity is that we need to calculate the area under the curve.

The total area under the normal curve is 1 (100 percent) and the area inside a given interval would give us the probability of the normally distributed random variable lying inside that range.

A confidence interval represents the range of values within which a population parameter is expected to lie for a specified percentage of the time. A 90 percent confidence interval for a return on the stock from -15 percent to 25 percent means that there is a 90 percent probability that the return on the stock will lie between -15 percent and 25 percent. Or there is 10 percent probability that the return will lie outside the range between -15 percent to 25 percent.

We usually don't have the values of population parameters. So, we estimate those value using sample statistics.

For a sample of a random variable with mean x ̅ and standard deviation, the confidence interval assuming the normal distribution are given below:

90 percent confidence interval is x ̅ ± 1.65s
95 percent confidence interval is x ̅ ± 1.96s
99 percent confidence interval is x ̅ ± 2.58s

It is advisable to remember the above confidence interval ranges. The approximation of the ranges is given below:

Approximately 50 percent of all observations lie in the interval x ̅ ± (2/3)s
Approximately 68 percent of all observations lie in the interval x ̅ ± 1.00s
Approximately 95 percent of all observations lie in the interval x ̅ ± 2.00s
Approximately 99 percent of all observations lie in the interval x ̅ ± 3.00s

Example 5: Calculating confidence interval

The annual return on a stock has a mean return of 10.00 percent and standard deviation of returns as 15.00 percent. Calculate the 95 percent confidence interval for the stock.

Solution:

The 95 percent confidence interval is calculated as x ̅ ± 1.96s

Therefore, 95 percent confidence interval for the returns on the stock = 10.00 percent ± 1.96*15.00 percent = -19.40 percent to 39.40 percent.

The returns on the stock is expected to lie between -19.40 percent and 39.40 percent for 95 percent of the time.



We can also calculate the probability that the return of a stock is positive, negative, greater than risk-free rate, greater than some percentage, smaller than some percentage, and in between some percentages. We need to look at the normal distribution probability distribution table to calculate those values. We also need to standardize our distribution to the standard normal distribution. We will discuss that in the next LOS.

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Next LOS: Standard normal distribution and standardizing a random variable

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