18  Distributions of Transformations of Random Variables

• A function of a random variable is a random variable: if $X$ is a random variable and $g$ is a function then $Y=g(X)$ is a random variable.
• Since $g(X)$ is a random variable it has a distribution. In general, the distribution of $g(X)$ will have a different shape than the distribution of $X$.

18.1 Linear rescaling

Example 18.1

Let $U$ have a Uniform(0, 1) distribution. Define $V=1-U$.

1. Does $V$ result from a linear rescaling of $U$?
   
   
   
2. What are the possible values of $V$?
   
   
   
3. Is $V$ the same random variable as $U$?
   
   
   
4. Find $\text{P}(U\le 0.1)$ and $\text{P}(V\le 0.1)$.
   
   
   
5. Sketch a plot of what the histogram of many simulated values of $V$ would look like.
   
   
   
6. Does $V$ have the same distribution as $U$?
   
   
   

Example 18.2

Suppose that $X$, the SAT Math score of a randomly selected student, follows a Normal(500, 100) distribution. Randomly select a student and let $X$ be the student’s SAT Math score. Now have the selected student spin the Normal(0, 1) spinner. Let $Z$ be the result of the spin and let $Y=500+100Z$.

1. Is $Y$ the same random variable as $X$?
   
   
   
2. Does $Y$ have the same distribution as $X$?
   
   
   

Example 18.3

Let $Z$ be a random variable with a Normal(0, 1) distribution. Consider $-Z$.

1. Donny Don’t says that the distribution of $-Z$ will look like an “upside-down bell”. Is Donny correct? If not, explain why not and describe the distribution of $-Z$.
   
   
   
2. Donny Don’t says that the standard deviation of $-Z$ is -1. Is Donny correct? If not, explain why not and determine the standard deviation of $-Z$.
   
   
   

• A linear rescaling is a transformation of the form $g(u)=a+bu$.
• A linear rescaling of a random variable does not change the basic shape of its distribution, just the range of possible values.
  • However, remember that the possible values are part of the distribution. So a linear rescaling does technically change the distribution, even if the basic shape is the same. (For example, Normal(500, 100) and Normal(0, 1) are two different distributions.)
• A linear rescaling transforms the mean in the same way the individual values are transformed.
• Adding a constant to a random variable does not affect its standard deviation.
• Multiplying a random variable by a constant multiplies its standard deviation by the absolute value of the constant.
• Whether in the short run or the long run, $$\begin{array}{rl}\text{Average of }a+bX& =a+b(\text{Average of }X)\\ \text{SD of }a+bX& =|b|(\text{SD of }X)\\ \text{Variance of }a+bX& =b^2(\text{Variance of }X)\end{array}$$
• If $U$ has a Uniform(0, 1) distribution then $X=a+(b-a)U$ has a Uniform($a$, $b$) distribution.
• If $Z$ has a Normal(0, 1) distribution then $X=\mu +\sigma Z$ has a Normal($\mu$, $\sigma$) distribution.
• Remember, do NOT confuse a random variable with its distribution.
  • The random variable is the numerical quantity being measured
  • The distribution is the long run pattern of variation of many observed values of the random variable

18.2 Nonlinear transformations of random variables

• In general the shape of the distribution $g(X)$ will be different than that of $X$.
• In general: whether in the short run or the long run

$$\text{Average of }g(X)\ne g(\text{Average of }X).$$

Example 18.4

Let $U$ have a Uniform(0, 1) distribution, and let $X=-\log (1-U)$. For each of the intervals in the table below find the probability that $U$ lies in the interval, and identify the corresponding values of $X$. (You should at least compute a few by hand to see what’s happening, but you can use software to fill in the rest.) After completing the table, sketch a histogram representing the distribution of $X$.

Interval of U	Length of U interval	Probability	Interval of X	Length of X interval
(0, 0.1)
(0.1, 0.2)
(0.2, 0.3)
(0.3, 0.4)
(0.4, 0.5)
(0.5, 0.6)
(0.6, 0.7)
(0.7, 0.8)
(0.8, 0.9)
(0.9, 1)

Example 18.5

For each of the intervals of $X$ values in the table below identify the corresponding values of $U$, and then find the probability that $X$ lies in the interval. (You should at least compute a few by hand to see what’s happening, but you can use software to fill in the rest.) After completing the table, sketch a histogram representing the distribution of $X$. Hint: if $X=-\log (1-U)$ then $U=1-e^{-X}$.

Interval of X	Length of X interval	Probability	Interval of U	Length of U interval
(0, 0.5)
(0.5, 1)
(1, 1.5)
(1.5, 2)
(2, 2.5)
(2.5, 3)
(3, 3.5)
(3.5, 4)
(4, 4.5)
(4.5, 5)

Example 18.6

We have now seen a few reasons why if $U$ has a Uniform(0, 1) distribution then $X=-\log (1-U)$ has an Exponential(1) distribution. The purpose of the example is derive the pdf of $X$, starting from just the setup in the previous sentence.

1. Identify the possible values of $X$. (We have done this already, but this should always be your first step.)
   
   
   
   
   
2. Let $F_X$ denote the cdf of $X$. Find $F_X(1)$.
   
   
   
   
   
3. Find $F_X(2)$.
   
   
   
   
   
4. Find the cdf $F_X(x)$.
   
   
   
   
   
5. Find the pdf $f_X(x)$.
   
   
   
   
   
6. Why should we not be surprised that $X=-\log (1-U)$ has cdf $F_X(x)=1-e^{-x}$? Hint: what is the function $u\mapsto -\log (1-u)$ in this case?
   
   
   
   
   

• If $X$ is a continuous random variable whose distribution is known, the cdf method can be used to find the pdf of $Y=g(X)$
• Determine the possible values of $Y$. Let $y$ represent a generic possible value of $Y$.
• The cdf of $Y$ is $F_Y(y)=\text{P}(Y\le y)=\text{P}(g(X)\le y)$.
• Rearrange $\{g(X)\le y\}$ to get an event involving $X$. Warning: it is not always $\{X\le g^{-1}(y)\}$. Sketching a picture of the function $g$ helps.
• Obtain an expression for the cdf of $Y$ which involves $F_X$ and some transformation of the value $y$.
• Differentiate the expression for $F_Y(y)$ with respect to $y$, and use what is known about $F_X^{\prime }=f_X$, to obtain the pdf of $Y$. You will typically need to apply the chain rule when differentiating.
• You will need to use information about $X$ at some point in the last step above. You can either:
  • Plug in the cdf of $X$ and then differentiate with respect to $y$.
  • Differentiate with respect to $y$ and then plug in the pdf of $X$.
  • Either way gets you to the correct answer, but depending on the problem one way might be easier than the other.

Example 18.7

Let $X$ be a random variable with a Uniform(-1, 1) distribution and Let $Y=X^2$.

1. Identify the possible values of $Y$.
   
   
   
   
   
2. Sketch the pdf of $Y$. Hint: consider a few equally spaced intervals of $Y$ values and see what $X$ values they correspond to.
   
   
   
   
   
3. Run a simulation to approximate the pdf of $Y$.
   
   
   
   
   
4. Find $F_Y(0.49)$.
   
   
   
   
   
5. Use the cdf method to find the pdf of $Y$. Is the pdf consistent with your simulation results?