Probability Density Function

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

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

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 Density Function

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.

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 2

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 3

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 $X$?