Question

If $$ \Phi(z)=\int_{-\infty}^{z} \frac{1}{\sqrt{2 \pi}} e^{-w^{2} / 2} d w $$ show that \(\Phi(-z)=1-\Phi(z)\)

Short answer

From the calculation, it is shown under these conditions that if \(\Phi(z)=\int_{-\infty}^{z} \frac{1}{\sqrt{2 \pi}} e^{-w^{2} / 2} d w\), then indeed \(\Phi(-z)=1-\Phi(z)\).

Step-by-step solution

Observe the given integral

First, examine the \(\Phi(z)\) and observe the relationship between \(\Phi(-z)\) and \(\Phi(z)\). The integral from \(-\infty\) to \(z\) represents the area under the curve of the standard normal distribution from \(- \infty\) up to \(z\). Due to the nature of the normal distribution, the area on either side of the zero mean is symmetrical. Therefore, the area from \(-\infty\) to \(-z\) should be equal to the area from \(z\) to \(\infty\). Knowing this symmetrical property will guide us in the next steps.

Change the integral limits

Express \(\Phi(-z)\) as an integral from \(z\) to \(\infty\) instead of \(-\infty\) to \(-z\). This is accomplished by the rule of changing limits which states that if the limits of a definite integral are interchanged, the value of the new integral is the negative of the old. By applying this rule, \(\Phi(-z)\) can be rewritten as \(-\int_{-\infty}^{-z} \frac{1}{\sqrt{2 \pi}} e^{-w^{2} / 2} d w\).

Apply properties of standard normal distribution

Now, remember the standard normal distribution property that the total probability under the curve is 1. This can be mathematically represented as \(\int_{-\infty}^{\infty}\frac{1}{\sqrt{2 \pi}} e^{-w^{2} / 2} d w = 1\). Then, split the integral by the middle, which is \(z\), leading to the formula \(\int_{z}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{-w^{2} / 2} d w + \int_{-\infty}^{z} \frac{1}{\sqrt{2 \pi}} e^{-w^{2} / 2} d w = 1\). By substitution, this formula becomes \(\Phi(-z)+\Phi(z)=1\).

Formulate the final equation

The final step is to rearrange the equation to prove the needful. Thus, \(\Phi(-z)=1-\Phi(z)\).

Key concepts

Normal Distribution Properties

The normal distribution, also known as the Gaussian distribution, plays a crucial role in statistics and is often used to represent real-valued random variables with unknown distributions in various fields, such as social sciences, finance, and natural sciences. One of its most defining features is its symmetric bell-shaped curve centered around the mean.

Some key properties include:

• Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal, reflecting its symmetry.
• Asymptotic: The tails of the curve approach the x-axis but never touch it, implying that the distribution has no real bounds.
• 68-95-99.7 Rule: Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

When dealing with the standard normal distribution specifically, the mean is 0 and the standard deviation is 1, simplifying many probabilistic calculations. The function \( \) represents the cumulative distribution function (CDF) for any z-value along the standard normal curve, quantifying the probability that a random variable will be less than or equal to z.

Definite Integral Manipulation

Definite integral manipulation is a mathematical technique used to solve problems involving the area under a curve or the accumulation of a quantity. It requires an understanding of the properties of integrals and the function concerned.

A fundamental theorem of calculus states that if F is an antiderivative of f over an interval, then:\[ \int_{a}^{b} f(x) dx = F(b) - F(a) \]For the normal distribution, we often need to deal with infinite bounds and use properties specific to the exponential function involved in its probability density function (PDF).

One such manipulation involves changing limits of integration, which can affect the sign of the resulting integral. For example, if you reverse the limits of an integral, the result is the negative of the original integral, as shown in the problem's solution step when converting \(\Phi(-z)\) to an integral from z to infinity. Another useful manipulation is splitting the integral into segments, aiding in the understanding of cumulative probabilities within different intervals of the distribution.

Symmetrical Probability Distribution

A symmetrical probability distribution is characterized by a balance about a central point, where each side of the distribution is a mirror image of the other. The standard normal distribution is a prime example of such symmetry, with the mean serving as the axis of symmetry.

Probabilities in a symmetrical distribution can be computed in various ways, and symmetry plays a key role in simplifying calculations. For instance, the probability of a random variable falling between two points equidistant from the mean is the same for both intervals.

The significance of this property is illustrated in the provided exercise, demonstrating that \(\Phi(z)\), representing the area to the left of z, and \(\Phi(-z)\), representing the area to the right of -z (and thus the complement of the area to the left of z), are symmetrically related. The proof \(\Phi(-z) = 1 - \Phi(z)\) showcases how symmetry can be exploited algebraically to deduce relationships between probabilities on opposite sides of the mean in a standard normal distribution.