Continuous PDFs, Interval Probabilities, and CDFs

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What is a probability density function?

A probability density function (pdf), written $f(x)$, is nonnegative and has total area 1. Areas under it represent probabilities over intervals.

How do you find an interval probability for the uniform density on $[0,20]$?

Multiply the interval width by the constant height $\frac{1}{20}$. For $0\leq a<b\leq20$, $P(a<X<b)=\frac{b-a}{20}$.

Why does a continuous random variable have zero probability at one exact value?

One value has zero width on the horizontal axis, so the area associated with it is zero: $P(X=c)=0$.

What does the cumulative distribution function $F(x)$ represent?

The CDF is $F(x)=P(X\leq x)$, the accumulated area under the pdf to the left of $x$.

How do you calculate a right-tail probability using the CDF?

Use $P(X>x)=1-F(x)$, since the areas to the left and right together total 1.

Quiz questions

1. Which two conditions must a probability density function satisfy?Answer: It is nonnegative, and its total area equals 1.
2. For $f(x)=\frac{1}{20}$ on $0\leq x\leq20$, what is the total area under the curve?Answer: $20\left(\frac{1}{20}\right)=1$
3. For the uniform density $f(x)=\frac{1}{20}$ on $[0,20]$, what is $P(4<X<15)$?Answer: $(15-4)\left(\frac{1}{20}\right)=0.55$
4. Why do $P(0<X<2)$ and $P(0\leq X\leq2)$ have the same value for a continuous random variable?Answer: The endpoints have zero probability, so including them adds no probability.
5. If $F(2)=P(X\leq2)=0.1$, what is $P(X>2)$?Answer: $1-F(2)=1-0.1=0.9$