2.1 Analytically – Part 2
2.1.2 Continuous Random Variable
While it is easier to grasp the concepts of mean and standard deviation using discrete distributions, many variables in economics such as stock returns and insurance losses are continuous random variables. The most common distribution for a continuous random variable is the normal distribution.
The density function of a normally distributed variable [latex]X[/latex] is depicted in Figure 2. The curve is bell-shaped and symmetric around the mean. For normal distribution, the mean is also the mode, the value that occurs most frequently, and the median, the number in the middle of a sorted list of numbers. The smaller the standard deviation, the more closely the possible outcomes will cluster about the mean. The higher the standard deviation, the more the possible outcomes are spread around the expected value, and as a result you are more likely to obtain an outcome that is very different from the one you expect.
A normally distributed random variable can take on any value from negative to positive infinity. However, there are certain established principles: 1. the probability that a normally distributed variable falls within one standard deviation from the mean is about 0.68, 2. the probability that it falls within two standard deviations from the mean is about 0.95, and 3. nearly the entire probability mass lies within three standard deviations from the mean.
Remember that for a discrete random variable the probabilities of all possible states should add up to one. For a continuous random variable, the area under the probability density curve should always be equal to one, regardless of the shape of the distribution.
Figure 2: Normal probability density function with mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex]
Example 4: Normal Distribution
Suppose it is reasonable to assume that the annual returns on another hypothetical biotech company, BioTech-X, stock follow the normal distribution depicted in Figure 3. As a result, the probability of getting a return in the range from 1% to 9% is 68.3%, and the probability of getting a return between -3% and 13% is 95.4%. Nearly 100% of the observed returns will lie between -7% and 17%. The probability of obtaining a return greater than 13% or lower than -3% is (100% – 95.4%) = 4.6%. As the normal distribution is symmetric, the probability of obtaining a return lower than -3% is equal to the probability of obtaining a return greater than 13%. If we are interested in the probability of obtaining extreme negative returns, we can compute the probability of getting returns lower than -3%, which is equal to 2.3% (4.6%/2).
![Figure 3: Annual returns of BioTech-X stock assuming normal distribution with mean [latex]\mu = 5[/latex] and standard deviation [latex]\sigma_{x} = 4[/latex] [NewTab]](https://ecampusontario.pressbooks.pub/app/uploads/sites/2426/2022/02/RISC-mod1-figure3.png)
Example 5: Standard Deviation and Risk
Compare the distributions of variables [latex]X[/latex] and [latex]Y[/latex] in Figure 4. The variables are both normally distributed and have the same mean, 5, but different standard deviations. Suppose these are the rates of return on two stocks, [latex]X[/latex] and [latex]Y[/latex], measured in %. Which one is riskier?
As the two variables have the same mean, the one with the higher standard deviation is riskier, i.e., stock [latex]Y[/latex].
![Figure 4: Two normal distributions with mean [latex]\mu = 5[/latex] and standard deviation [latex]\sigma_{x} = 4[/latex] and [latex]\sigma_{y} = 6[/latex] [NewTab]](https://ecampusontario.pressbooks.pub/app/uploads/sites/2426/2022/02/RISC-mod1-figure4-two-normal-with-different-std.jpg)
A random variable whose outcome can take on any real value in an interval.
