21: Change of Variables

Author

Derek Sollberger

Published

March 15, 2023

Linear Conversion

Let \(F\) be the daily high temperature in Fahrenheit in Merced, California, with a mean of 76 degrees and a standard deviation of 15 degrees. Compute those sample statistics in Celsius.

Temperature Conversion

We know that the conversion formula is

\[C = \displaystyle\frac{5}{9}(F - 32)\]

Range Rule of Thumb

Range Rule of Thumb

Recall

  • About 67 percent of data falls within one standard deviation of the mean
  • About 95 percent of data falls within two standard deviations of the mean

\[\left( \mu - 2\sigma, \mu + 2\sigma \right)\]

We had computed

  • \(\mu_{F} \approx 76\) and \(\sigma_{F} \approx 15\) degrees Fahrenheit
  • \(\mu_{C} \approx 24.4444\) and \(\sigma_{C} \approx 8.3333\) degrees Celsius

Build range-rule-of-thumb intervals for the Merced high temperatures in Fahrenheit and in Celsius.

Distributions

Determine the distribution and density functions for

\[Y = \displaystyle\frac{5}{9}(X - 32)\]

Change of Coordinates

Change of Coordinates

Let \(X\) be a continuous random variable with distribution function \(F_{X}\) and density function \(f_{X}\). If we apply a linear transformation

\[Y = aX + c\]

where \(a >0\) and \(c\) are constants, then

\[F_{Y}(y) = F_{X}\left(\displaystyle\frac{y - c}{a}\right) \text{ and } f_{Y}(y) = \displaystyle\frac{1}{a}f_{X}\left(\displaystyle\frac{y - c}{a}\right)\]

If \(X \sim Exp(1/2)\), then what kind of distribution does \(Y = 32X\) have?

Nonlinear Transformations

Let \(X \sim U\left(0, \displaystyle\frac{\pi}{2}\right)\) and \(Y = \sin(X)\).

Compare \(\text{E}[\sin X]\) and \(\sin(\text{E}[X])\)

Suppose that a disease outbreak can be modeled where \(X\) is the population density of a city and \(Y\) is the number of diagnosed cases with

\[X \sim U(0,100), \quad Y = X^{3.2}\]

Compare \(\text{E}[X^{3.2}]\) and \(\left(\text{E}[X]\right)^{3.2}\)

Jensen’s Inequality

The previous two examples were demonstrations of , which states that

  • If \(g\) is a convex function of random variable \(X\), then

\[g(\text{E}[X]) \leq \text{E}[g(X)]\] - If \(g\) is a concave function of random variable \(X\), then

\[g(\text{E}[X]) \geq \text{E}[g(X)]\]

where the equal signs are not included when the function \(g\) is strictly convex or strictly concave.

Looking Ahead

  • due Fri., Mar. 17:

    • WHW8
    • LHW7

  • no lecture on Mar. 24, Apr. 3

  • Exam 2 will be on Mon., Apr. 10

Misc