Continuous Random Variables

A continuous random variable is a variable whose value can take on any value within a given range or interval.

Probability Density Functions (PDFs)

Definition

A probability density function (PDF), denoted by $f(x)$, is a function that describes the relative likelihood for a continuous random variable to take on a given value.

Properties of a PDF

1. Non-negativity: $f(x) \geq 0$ for all $x$.
2. Total Area: The total area under the curve of $f(x)$ over its entire range is equal to 1. Mathematically, $\int_{-\infty}^{\infty} f(x) \, dx = 1$.

Construction of PDFs

PDFs are constructed from non-negative functions of a real variable. To make a non-negative function a valid PDF, it must be scaled such that the total area under the curve is equal to 1. This often involves finding a constant $k$ such that $\int_{-\infty}^{\infty} k \cdot g(x) \, dx = 1$, where $g(x)$ is the original non-negative function.

KEY TAKEAWAY: A PDF must be non-negative and have a total area under the curve of 1.

Specification of Probability Distributions

A probability distribution for a continuous random variable is specified by its PDF, $f(x)$. This function allows us to calculate probabilities associated with the random variable.

Mean, Variance, and Standard Deviation

Mean ($\mu$)

The mean (or expected value) of a continuous random variable $X$ with PDF $f(x)$ is given by:

$$\mu = E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$$

Variance ($\sigma^2$)

The variance of a continuous random variable $X$ with PDF $f(x)$ and mean $\mu$ is given by:

$$\sigma^2 = Var(X) = E[(X - \mu)^2] = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot f(x) \, dx$$

A more convenient formula for calculation is:

$$\sigma^2 = E(X^2) - \mu^2 = \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx - \mu^2$$

Standard Deviation ($\sigma$)

The standard deviation is the square root of the variance:

$$\sigma = \sqrt{Var(X)}$$

Interpretation and Use

• The mean represents the average value of the random variable.
• The variance and standard deviation measure the spread or dispersion of the distribution around the mean. A higher variance/standard deviation indicates greater variability.

EXAM TIP: Remember to use the correct formulas for mean and variance. Pay attention to the integration limits based on the PDF’s domain.

Standard Normal Distribution and Transformed Normal Distributions

Standard Normal Distribution

The standard normal distribution, denoted by $N(0, 1)$, is a normal distribution with a mean of 0 and a standard deviation of 1. Its PDF is given by:

$$f(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}$$

where $z$ is the standard normal variable.

Transformed Normal Distribution

A transformed normal distribution, denoted by $N(\mu, \sigma^2)$, is a normal distribution with a mean of $\mu$ and a variance of $\sigma^2$. Its PDF is given by:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$

where $x$ is the random variable.

Standardization

Any normal distribution $N(\mu, \sigma^2)$ can be transformed into the standard normal distribution $N(0, 1)$ by using the z-score formula:

$$z = \frac{x - \mu}{\sigma}$$

This allows us to use standard normal tables or calculators to find probabilities associated with any normal distribution.

REMEMBER: The standard normal distribution is a special case of the normal distribution with $\mu=0$ and $\sigma=1$.

Effect of Parameters on the Graph of a PDF

Normal Distribution Parameters

For a normal distribution $N(\mu, \sigma^2)$:

• Mean ($\mu$): Changing the mean shifts the graph horizontally. Increasing $\mu$ shifts the graph to the right, while decreasing $\mu$ shifts it to the left. The shape remains the same.
• Standard Deviation ($\sigma$): Changing the standard deviation affects the spread of the graph. Increasing $\sigma$ makes the graph wider and flatter, while decreasing $\sigma$ makes it narrower and taller. The area under the curve remains 1.

General PDF Parameters

For a general PDF, the effect of changing parameters depends on the specific function. However, common effects include:

• Scaling parameters: Affect the height and width of the graph.
• Translation parameters: Shift the graph horizontally or vertically.
• Shape parameters: Alter the overall shape of the distribution.

VCAA FOCUS: VCAA often tests your understanding of how changing parameters affects the shape and position of a normal distribution.

Calculation of Probabilities

Probability as Area

The probability that a continuous random variable $X$ lies within an interval $(a, b)$ is given by the area under the PDF $f(x)$ between $a$ and $b$:

$$P(a < X < b) = \int_{a}^{b} f(x) \, dx$$

Cumulative Distribution Function (CDF) - Not Required

The cumulative distribution function (CDF), denoted by $F(x)$, gives the probability that the random variable $X$ is less than or equal to $x$:

$$F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \, dt$$

Although the CDF is not required knowledge, it can be a useful tool. Then, $P(a < X < b) = F(b) - F(a)$.

Conditional Probability

The conditional probability of event A given event B is:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

For continuous random variables, this often involves finding the probability of $X$ being in a specific interval given that it is already in another interval. For example:

$$P(a < X < b | c < X < d) = \frac{P(a < X < b \cap c < X < d)}{P(c < X < d)}$$

Where $a < b$ and $c < d$. If the intervals overlap, adjust the intersection accordingly.

If $c < a < b < d$, then

$$P(a < X < b | c < X < d) = \frac{P(a < X < b)}{P(c < X < d)} = \frac{\int_{a}^{b} f(x) \, dx}{\int_{c}^{d} f(x) \, dx}$$

COMMON MISTAKE: When calculating conditional probabilities, make sure to correctly identify the intersection of the events and divide by the probability of the condition.

Using Technology

Calculators can be used to evaluate definite integrals and find probabilities associated with continuous random variables. Specifically, use the normal CDF function for normal distributions.

APPLICATION: Continuous random variables and their PDFs are used in a wide variety of fields, including engineering, finance, and physics, for modeling and analyzing data.