The Normal Distribution (VCE SSCE Mathematical Methods): Revision Notes

The Normal Distribution

The standard normal distribution

The most basic version of the normal distribution is called the standard normal distribution. This special distribution has a probability density function defined by:

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

where the domain is all real numbers ($x \in \mathbb{R}$).

The graph of this function creates the familiar bell-shaped curve that is symmetric around the vertical line $x = 0$. This symmetry exists because $f(-x) = f(x)$, making it an even function.

Key features of the standard normal distribution curve:

• The line $y = 0$ acts as a horizontal asymptote: as $x$ approaches $\pm\infty$, the curve approaches but never touches the $x$-axis
• Almost all of the area under the curve lies between $x = -3$ and $x = 3$
• The curve is perfectly symmetric around $x = 0$
• The total area under the curve equals 1 (as with all probability density functions)

Mean and standard deviation of the standard normal distribution

From the symmetry of the graph, we can see that both the mean and median equal 0. The curve is centered at the origin.

Although we cannot find an exact integral for the standard normal probability density function, the symmetry allows us to verify that the mean is 0. When we split the integral for $E(X)$ into two parts (from $-\infty$ to 0 and from 0 to $\infty$), one integral is simply the negative of the other, so they cancel out.

For the variance and standard deviation, it can be shown through integration that:

$$E(X^2) = \int_{-\infty}^{\infty} x^2 f(x) \, dx = 1$$

Therefore:

$$\text{Var}(X) = E(X^2) - [E(X)]^2 = 1 - 0 = 1$$ $$\text{sd}(X) = \sqrt{\text{Var}(X)} = 1$$

The general normal distribution

The normal distribution is not limited to cases where the mean is 0 and standard deviation is 1. We can create normal distributions with any mean μ and any positive standard deviation σ.

Transformations of the standard normal distribution

To obtain a normal distribution with mean $\mu$ and standard deviation $\sigma$ from the standard normal distribution, we apply the following transformation:

$$(x, y) \rightarrow \left(\sigma x + \mu, \frac{y}{\sigma}\right)$$

This transformation consists of:

• A dilation by factor $\sigma$ from the $y$-axis
• A dilation by factor $\frac{1}{\sigma}$ from the $x$-axis
• A translation by $\mu$ units in the positive $x$-direction (when $\mu > 0$)

Going in the reverse direction, we can transform from a general normal distribution back to the standard normal using:

$$(x, y) \rightarrow \left(\frac{x - \mu}{\sigma}, \sigma y\right)$$

This transformation:

• Translates by $\mu$ units in the negative $x$-direction
• Dilates by factor $\frac{1}{\sigma}$ from the $y$-axis
• Dilates by factor $\sigma$ from the $x$-axis

Area-preserving property

For example, with $\mu = 100$ and $\sigma = 15$, a rectangle with area 180 square units in the general normal distribution corresponds to a rectangle with the same area in the standard normal distribution, just with different dimensions.

The general normal distribution formula

If $X$ is a normally distributed random variable with mean $\mu$ and standard deviation $\sigma$, then its probability density function is:

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

To find probabilities, we use the relationship:

$$\Pr(X \leq a) = \Pr\left(Z \leq \frac{a - \mu}{\sigma}\right)$$

where $Z$ is the standard normal random variable.

Properties of the normal distribution

All normal distributions share certain fundamental properties:

• Probability equals area: The probability of an event corresponds to the area under the curve
• Total area is 1: The total area under any normal curve equals 1
• Bell-shaped: The curve has a characteristic bell shape
• Symmetric: The curve is symmetric around its mean

Effect of the mean ($\mu$)

The mean determines the location (center) of the curve. The curve is symmetric about the vertical line $x = \mu$.

When we compare two normal distributions with the same standard deviation but different means:

• The shapes are identical
• The curves are centered at different positions along the $x$-axis
• If $\mu_1 < \mu_2$, then the first curve is to the left of the second

Effect of the standard deviation ($\sigma$)

The standard deviation determines the spread (width) of the curve.

Key points about the effect of $\sigma$:

• The curve has a maximum value of $\frac{1}{\sigma\sqrt{2\pi}}$ at $x = \mu$
• Larger σ means a wider, flatter curve
• Smaller σ means a taller, narrower curve
• When comparing distributions with the same mean but different standard deviations, all curves are centered at the same point but have different spreads

Consistent areas within standard deviations

Worked example

Remember!