Lecture 22

Qi Wang, Department of Statistics

Oct 15, 2018

**Definition:**a random variable that can take on any value in a range.- For example:
- Blood pressure
- The height of a 8 year old
- Mile per gallon of a car
- GPA

**Definition:**a function that gives the probability that a**discrete**random variable is exactly equal to some value.- For every x, $0 \le p_X(x) \le 1$
- $\sum_x{p_X(x)} = 1$
- For example, $X \sim Binomial(n = 4, p = 0.3)$, $P(X \le 2) = P_X(0) + P_X(1) + P_X(2)$

In order to find probabilities of continuous random variables, we can no longer use a PMF, because the probabilities are no longer at points, they are over regions. Instead we have a Probability Density Function, or PDF, $f_X(x)$, which looks more like a traditional function over a region. If you are given a PDF, the probability can be calculated as: $$P(X \le a) = \int_{-\infty}^{a} f_X(x) dx$$

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$
- $P(X \le a) = \int_{\infty}^{a} f_X(x) dx$
- $P(a \le X \le b) = \int_{a}^{b} f_X(x) dx$

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

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

The cumulative distribution function F(x) for a continuous random variable X is defined for every number x by $$F_X(x) = P(X \le x) = \int_{-\infty}^{x} f_X(t) dt$$ For each x, $F_(x)$ is the area under the density curve to the left of x.

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}

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

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.

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

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

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