In simple words, **The Q-function gives the probability that a random variable from a normal distribution will exceed a certain threshold value**. **The erf function gives the probability that a normally distributed variable will fall within a certain range**.

## Q function

Q functions are often encountered in the theoretical equations for Bit Error Rate (BER) involving AWGN channel. A brief discussion on Q function and its relation to ** erfc** function is given here.

Gaussian process is the underlying model for an AWGN channel.The probability density function of a Gaussian Distribution is given by

\[p(x) = \displaystyle{ \frac{1}{ \sigma \sqrt{2 \pi}} e^{ – \frac{(x-\mu)^2}{2 \sigma^2}}}\quad\quad (1) \]

Generally, in BER derivations, the probability that a Gaussian Random Variable \(X \sim N ( \mu, \sigma^2) \) exceeds \(x_0\) is evaluated as the area of the shaded region as shown in Figure 1.

Mathematically, the area of the shaded region is evaluated as,

\[Pr(X \geq x_0) =\displaystyle{ \int_{x_0}^{\infty} p(x) dx = \int_{x_0}^{\infty} \frac{1}{ \sigma \sqrt{2 \pi}} e^{ – \frac{(x-\mu)^2}{2 \sigma^2}} dx } \quad\quad (2) \]

The above probability density function given inside the above integral cannot be integrated in closed form. So by change of variables method, we substitute

\[\displaystyle{ y = \frac{x-\mu}{\sigma} }\]

Then equation (2) can be re-written as,

\[\displaystyle{ Pr\left( y > \frac{x_0-\mu}{\sigma} \right ) = \int_{ \left( \frac{x_{0} -\mu}{\sigma}\right)}^{\infty} \frac{1}{ \sqrt{2 \pi}} e^{- \frac{y^2}{2}} dy } \quad\quad (3) \]

Here the function inside the integral is a normalized gaussian probability density function \(Y \sim N( 0, 1)\), normalized to mean \(\mu=0\) and standard deviation \(\sigma=1\).

The integral on the right side can be termed as Q-function, which is given by,

\[\displaystyle{Q(z) = \int_{z}^{\infty}\frac{1}{ \sqrt{2 \pi}} e^{- \frac{y^2}{2}} dy } \quad\quad (4)\]

Here the Q function is related as,

\[\displaystyle{ Pr\left( y > \frac{x_0-\mu}{\sigma} \right ) = Q\left(\frac{x_0-\mu}{\sigma} \right ) = Q(z)} \quad\quad (5)\]

Thus Q function gives the area of the shaded curve with the transformation \(y = \frac{x-\mu}{\sigma}\) applied to the Gaussian probability density function. Essentially, Q function evaluates the tail probability of normal distribution (area of shaded area in the above figure).

**The Q-function gives the probability that a random variable from a normal distribution will exceed a certain threshold value**.

## Error function

The complementary error function represents the area under the two tails of zero mean Gaussian probability density function of variance \(\sigma^2 = 1/2\). The error function gives the probability that the parameter lies outside that range.

Therefore, the complementary error function is given by

\[\displaystyle{ erfc(z) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-x^2}} dx \quad\quad (6)\]

Hence, the error function is

\[erf(z) = 1 – erfc(z) \quad\quad (7)\]

or equivalently,

\[\displaystyle{ erf(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-x^2} dx } \quad\quad (8) \]

**The erf function gives the probability that a normally distributed variable will fall within a certain range**.

## Q function and Complementary Error Function (erfc)

From the limits of the integrals in equation (4) and (6) one can conclude that ** Q function** is directly related to complementary error function (

**).It follows from equation (4) and (6),**

*erfc***is related to complementary error function by the following relation.**

*Q function*\[\displaystyle{ Q(z) = \frac{1}{2} erfc \left( \frac{z}{\sqrt{2}}\right)} \quad\quad (9) \]

## Some important results

Keep a note of the following equations that can come handy when deriving probability of bit errors for various scenarios. These equations are compiled here for easy reference.

If we have a normal variable \(X \sim N (\mu, \sigma^2)\), the probability that \(X > x\) is

\[\displaystyle{ Pr \left( X > x \right) = Q \left( \frac{x-\mu}{\sigma} \right ) } \quad\quad (10) \]

If we want to know the probability that \(X\) is away from the mean by an amount** ‘a’ **(on the left or right side of the mean), then

\[\displaystyle{ Pr \left( X > \mu+a \right) = Pr \left( X < \mu-a \right) = Q\left(\frac{a}{\sigma} \right ) } \quad\quad (11) \]

If we want to know the probability that X is away from the mean by an amount * ‘a’* (on both sides of the mean), then

\[\displaystyle{ Pr \left( \mu-a > X > \mu+a \right) = 2 Q\left(\frac{a}{\sigma} \right ) } \quad\quad (12)\]

Application of Q function in computing the Bit Error Rate (BER) or probability of bit error will be the focus of our next article.

## Applications

The **Q-function** and the **error function (erf)** are important mathematical functions that arise in many fields, including probability theory, statistics, signal processing, and communications engineering. Here are some reasons why these functions are important:

**Probability calculations:**The Q-function and erf function are used in probability calculations involving Gaussian distributions.**The Q-function gives the probability that a random variable from a normal distribution will exceed a certain threshold value**.**The erf function gives the probability that a normally distributed variable will fall within a certain range**.- Signal processing: In signal processing, the
**Q-function is used to calculate the probability of bit error in digital communication systems**. This is important for designing communication systems that can reliably transmit data over noisy channels. - Statistical analysis: The Q-function and erf function are used in statistical analysis to model data and estimate parameters. For example, in hypothesis testing, the Q-function can be used to calculate p-values.
- Mathematical modeling: The Q-function and erf function arise naturally in mathematical models for various phenomena. For example, the heat equation in physics and the Black-Scholes equation in finance both involve the erf function.
- Computational efficiency: In some cases, the Q-function and erf function provide a more efficient and accurate way of calculating certain probabilities and integrals than other methods.

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