21: Change of Variables

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

$$(\mu -2\sigma ,\mu +2\sigma )$$

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

Concave

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

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

Convex

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

$$X\sim U(0,100),Y=X^{3.2}$$

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

Jensen’s Inequality

The previous two examples were demonstrations of , which states that

• If $g$ is a convex function of random variable $X$, then

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

$$g(\text{E}[X])\ge \text{E}[g(X)]$$

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