# Q function (gaussian random variable)

• magnifik
In summary, for X ~ N(μ, σ), the probability P[|X-μ| < σ] can be expressed as 1 - 2Q(1), where Q(1) represents the tail probability under the bell curve to the right of μ+σ. This can be deduced by graphically representing the bell curve and using its symmetry to calculate the total surface under the curve.

#### magnifik

For X ~ N(μ, σ), what is P[|X-μ] < σ] in terms of the Q function?

I know that P[|X-μ] < σ] can be decomposed into P[X > -σ + μ] + P[X < σ + μ] I'm not sure what to do next. i know P[X < σ + μ] can be expressed as 1 - phi(σ + μ - μ / σ) = Q(1), but I'm not sure how to approach P[X > -σ + μ]. I know the answer is 1 - 2Q(1)...not sure where this all comes from

Hi magnifik!

I'm afraid you have the wrong decomposition.

Things should become clearer when you look at them graphically.
Draw a bell curve with μ in the middle and standard deviation σ.

P[|X-μ| < σ] corresponds to the surface under the bell curve between μ-σ and μ+σ

Q(1) corresponds to the surface under the bell curve to the right of μ+σ (the tail probability).
Since the bell curve is symmetric, Q(1) also corresponds to the surface to the left of μ-σ.

Since the total surface under the bell curve is 1, you can deduce your formula.

## 1. What is the Q function for a gaussian random variable?

The Q function, also known as the tail probability of a gaussian random variable, is a mathematical function that measures the probability that a gaussian random variable will exceed a certain threshold. It is commonly used in signal processing and communication systems.

## 2. How is the Q function related to the cumulative distribution function (CDF) of a gaussian random variable?

The Q function is the complement of the CDF of a gaussian random variable. This means that the Q function calculates the probability of the random variable exceeding a certain threshold, while the CDF calculates the probability of it being less than or equal to the threshold.

## 3. Can the Q function be expressed in terms of other mathematical functions?

Yes, the Q function can be expressed in terms of the complementary error function (erfc) or the inverse error function (erfinv). These functions are commonly used to approximate the Q function for different values of the threshold and standard deviation.

## 4. What is the relationship between the Q function and the signal-to-noise ratio (SNR)?

The Q function is often used to analyze the performance of communication systems, where the SNR is a key parameter. The Q function can be used to calculate the bit error rate (BER) of a digital communication system, which is directly related to the SNR.

## 5. Are there any applications of the Q function outside of signal processing and communication systems?

Yes, the Q function has applications in various fields such as statistics, finance, and physics. In statistics, it is used to calculate the confidence interval for a normal distribution. In finance, it is used to model stock price movements. In physics, it is used to analyze the noise in electronic devices.