AppliedStatsCourse

Density Curves

Reading

• Section 2.5

Practice Problems

2.6.5 (page 125) ~ 2.43, 2.44

Notes

• The book refers to them as “continuous probability functions”.
• Think of them like “smooth histograms”.
• Used to describe large (infinite) populations.

• Key property:

  Area under the curve and between $v_1$, $v_2$ is equal to the percent of data values that are between $v_1$ and $v_2$.

  Total area under curve equals 1 (100%).

• To really compute things we would need to know the equation of this curve, and do some Calculus.
• Distributions have parameters that specify their exact shape.

• Some examples of distributions that are easy to work with:

  Uniform ~ Straight line from $a$ to $b$. Represents the idea that all numbers between $a$ and $b$ are equally likely. For example most computers are equipped with “random number generators” that produce uniformly random numbers between $0$ and $1$.

  Normal ~ Has well known formula and tables to use. We will see this in next section. Plays a dominant role in statistics because of the “Central Limit Theorem” we will discuss later.

Uniform Distribution

• Graph is a straight line at height $\frac{1}{b-a}$, extending from $x=a$ and $x=b$.

• Can compute areas as they are just rectangles.

  Shape ~ Symmetric

  Mean ~ $\frac{a+b}{2}$

  Median ~ $\frac{a+b}{2}$

  Std Dev ~ $\sqrt{\frac{(b-a)^2}{12}}$

  Quartiles ~ One fourth and three fourths of the way from $a$ to $b$.

  IQR ~ $\frac{b-a}{2}$

Example: Consider the uniform distribution from $a=1$ to $b=3$.

• Draw the graph.
• Compute using the graph the amount of data between $x=1.2$ and $x=2.9$.
• Find using the graph the location of the 90th percentile.
• Compute the mean, standard deviation, and IQR.