Introduction to Probability Models

Lecture 23

Qi Wang, Department of Statistics

Oct 18, 2017

Last Three Discrete Random Variables

  • Poisson
  • Geometric
  • Negative Binomial

Poisson Distribution

  • $X \sim Poisson(\lambda)$
  • The definition of $X$: the number of success per $\underline{\hspace{1cm}}$, and $\underline{\hspace{1cm}}$ can be time, length, space unit and so on
  • Support: $\{0, 1, 2, \cdots\}$
  • Parameters: $\lambda$, the average success rate per $\underline{\hspace{1cm}}$
  • PMF: $P_X(x) = \frac{e^{-\lambda} \lambda^x}{x!}$
  • Expected Value: $E[X] = \lambda$
  • Variance: $Var(X) = \lambda$

Geometric Distribution

  • $X \sim Geom(p)$
  • The definition of $X$ : the number of trials to get the first success
  • Support: $\{1, 2, \cdots\}$, NOTE: NO ZERO!
  • Parameter: $p$, the probability of success in one trial
  • PMF: $P_X(x) = p(1-p)^{x - 1}$
  • Expected Value: $E[X] = \frac{1}{p}$
  • Variance: $Var(X) = \frac{1 - p}{p^2}$
  • Tail Probability formula: $P(X > k) = (1 - p)^k$
  • Memoryless Property: $P(X > s + t| X > s) = P(X > t)$ and $P(X < s + t| X > s) = P(X < t)$

Negative Binomial Distribution

  • $X \sim NegBin(r, p)$ or $X \sim NB(r, p)$
  • The definition of $X$ : the number of trials to get the $r_{th}$ success
  • Support: $\{r, r + 1, r + 2, \cdots\}$
  • Parameter:
    • $p$: the probability of success in one trial
    • $r$: success of interest
  • PMF: $P_X(x) = C_{r-1}^{x-1}p^r(1-p)^{x - r}$
  • Expected Value: $E[X] = \frac{r}{p}$
  • Variance: $Var(X) = \frac{r(1 - p)}{p^2}$

Time for Quiz

Properties of a Continous PDF

If $f_X(x)$ is a probability density function(PDF), then

  • $f_X(x) \ge 0$ for all values of $X$ in its support
  • $ \int_{-\infty}^{\infty} f_X(x) dx = 1$
  • $P(a \le X \le b) = \int_{a}^{b} f_X(x) dx$

Example 1

Let $f_X(x) = 0.25 x$ for $1 \le x \le 3$ and $0$ otherwise

  1. Make of graph of the PDF
  2. Is $X$ more likely to be in the interval $[1, 2]$ or $[2, 3]$

Cumulative Distribution Function(CDF)

  • Definition: if $f_X(x)$ is PDF, then CDF: $F_X(x) = \int_{-\infty}^x f_X(t) dt$
  • $F_X(x) \ge 0$ for all values of $X$ in its support
  • $F_X(x)$ is non-decreasing
  • $F_X(-\infty) = 0$, $F_X(\infty) = 1$

Example 1 continued

  • Find the CDF $F_X(x)$
  • Find $P(X < 2.2)$ and $P(X > 1.7)$

Example 2

Let X represent the diameter in inches of a circular disk cut by a machine. Let $f_X(x) = c(4x - x^2)$ and $0$ otherwise.

  1. Find the value of c that makes this a valid PDF.
  2. Find the CDF $F_X(x)$
  3. Find the probability that X is within 0.5 inches of the average diameter.