Introduction to Probability Models

Lecture 24

Qi Wang, Department of Statistics

Oct 20, 2017

Important Concepts

  • PDF: $f_X(x)$ is a non-negative function of $x$ which integrates to 1, $$f_X(x) \ge 0, \int_{-\infty}^{\infty}f_X(x) dx = 1$$
  • CDF: The area under the PDF to the left of a point x, $$F_X(x) = P(X \le x) = \int_{-\infty}^{x}f_X(x) dx$$
  • Generally, $P(a \le X \le b) = \int_{a}^{b} f_X(x) dx = F_X(b) - F_X(a)$

Expected Value and Variance for a Continuous Random Variable

X is a continuous random variable with PDF $f_X(x)$

  • Expected Value: $$E[X] = \int_{-\infty}^{+\infty} x f_X(x) dx$$
  • Variance: \begin{align} Var(X) &= E[(E - E[X])^2] \\ & = E[X^2] - E[X]^2 \\ &= \int_{-\infty}^{+\infty} x^2f_X(x) dx - (\int_{-\infty}^{+\infty} xf_X(x) dx)^2 \end{align}

Percentiles

X is a continuous random variable with PDF $f_X(x)$ and CDF $F_X(x)$

  • Definition:$p_{th}$ percentile of X means a value k at which $P(X < k) = p\%$ or $F_X(k) = p\%$
  • For example, $10_{th}$ percentile of X means a value k at which $P(X < k) = 0.10$ or $F_X(k) = 0.10$
  • Quantile:
    • $1_{st}$ quantile: $25_{th}$ percentile
    • $2_{nd}$ quantile, Median: $50_{th}$ percentile
    • $3_{rd}$ quantile: $75_{th}$ percentile

Example 1

Let X represent the diameter in inches of a circular disk cut by a machine. Let $f_X(x) = c(4x - x^2)$ for $1 \le x \le 4$ and $0$ otherwise.

  1. Find the value of c that makes this a valid PDF.
  2. Find the expected value and variance of $X$
  3. Find the CDF $F_X(x)$
  4. Find the probability that X is within 0.5 inches of the average diameter.
  5. What is the third quartile of X?

Example 2

Suppose that a continuous random variable, X, has the probability density function (PDF) given below: \[ f_X(x)=\left\{ \begin{array}{ll} \frac{3}{2}x, 0 \le x \le 1\\ \frac{1}{4}, 5 \le x \le 6\\ 0, otherwise \end{array} \right. \]

  1. What is the probability that $X$ is equal to 4?
  2. What is the probability that $X$ is more than 4?
  3. Find $F_X(5.6)$
  4. Knowing that $X$ is more than $0.8$, what is the probability that $X$ is less than 5.6?
  5. What is the $85_{th}$ percentile of $X$?