Random Variables & Stochastic Processes

For a full treatment of random variables and stochastic processes (sequences of random variables), see, e.g., [201]. For practical every-day signal analysis, the simplified definitions and examples below will suffice for our purposes.

Probability Distribution


Definition: A probability distribution $ \hat{p}(x)$ may be defined as a non-negative real function of all possible outcomes of some random event. The sum of the probabilities of all possible outcomes is defined as 1, and probabilities can never be negative.


Example: A coin toss has two outcomes, ``heads'' (H) or ``tails'' (T), which are equally likely if the coin is ``fair''. In this case, the probability distribution is

$\displaystyle \hat{p}(H) = \hat{p}(T) = \frac{1}{2}$ (C.1)

where $ \hat{p}(x)$ denotes the probability of outcome $ x$ . That is, the total ``probability mass'' is divided equally between the two possible outcomes heads or tails. This is an example of a discrete probability distribution because all probability is assigned to two discrete points, as opposed to some continuum of possibilities.


Independent Events

Two probabilistic events $ H_1$ and $ H_2$ are said to be independent if the probability of $ H_1$ and $ H_2$ occurring together equals the product of the probabilities of $ H_1$ and $ H_2$ individually, i.e.,

$\displaystyle \hat{p}(H_1 H_2) = \hat{p}(H_1)(H_2)$ (C.2)

where $ \hat{p}(H_1 H_2)$ denotes the probability of $ H_1$ and $ H_2$ occurring together.


Example: Successive coin tosses are normally independent. Therefore, the probability of getting heads twice in a row is given by

$\displaystyle \hat{p}(H H) = \hat{p}(H)\hat{p}(H) = \frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}.$ (C.3)


Random Variable


Definition: A random variable $ x$ is defined as a real- or complex-valued function of some random event, and is fully characterized by its probability distribution.


Example: A random variable can be defined based on a coin toss by defining numerical values for heads and tails. For example, we may assign 0 to tails and 1 to heads. The probability distribution for this random variable is then

$\displaystyle \hat{p}(x) = \left\{\begin{array}{ll} \frac{1}{2}, & x = 0 \\ [5pt] \frac{1}{2}, & x = 1 \\ [5pt] 0, & \mbox{otherwise}. \\ \end{array} \right. \protect$ (C.4)


Example: A die can be used to generate integer-valued random variables between 1 and 6. Rolling the die provides an underlying random event. The probability distribution of a fair die is the discrete uniform distribution between 1 and 6. I.e.,

$\displaystyle \hat{p}(x) = \left\{\begin{array}{ll} \frac{1}{6}, & x = 1,2,\ldots,6 \\ [5pt] 0, & \mbox{otherwise}. \\ \end{array} \right.$ (C.5)


Example: A pair of dice can be used to generate integer-valued random variables between 2 and 12. Rolling the dice provides an underlying random event. The probability distribution of two fair dice is given by

$\displaystyle \hat{p}(x) = \left\{\begin{array}{ll} \frac{x-1}{36}, & x = 2,3,\ldots,7 \\ [5pt] \frac{13-x}{36}, & x = 7,8,\ldots,12 \\ [5pt] 0, & \mbox{otherwise}. \\ \end{array} \right.$ (C.6)

This may be called a discrete triangular distribution. It can be shown to be given by the convolution of the discrete uniform distribution for one die with itself. This is a general fact for sums of random variables (the distribution of the sum equals the convolution of the component distributions).


Example: Consider a random experiment in which a sewing needle is dropped onto the ground from a high altitude. For each such event, the angle of the needle with respect to north is measured. A reasonable model for the distribution of angles (neglecting the earth's magnetic field) is the continuous uniform distribution on $ [0,2\pi)$ , i.e., for any real numbers $ a$ and $ b$ in the interval $ [0,2\pi)$ , with $ a\leq
b$ , the probability of the needle angle falling within that interval is

$\displaystyle \int_a^b \frac{1}{2\pi}d\theta = \frac{1}{2\pi}(b-a), \quad a,b\in[0,2\pi).$ (C.7)

Note, however, that the probability of any single angle $ \theta$ is zero. This is our first example of a continuous probability distribution. Therefore, we cannot simply define the probability of outcome $ \theta$ for each $ \theta\in [0,2\pi)$ . Instead, we must define the probability density function (PDF):

$\displaystyle p(\theta) = \left\{\begin{array}{ll} \frac{1}{2\pi}, & 0\leq \theta < 2\pi \\ [5pt] 0, & \mbox{otherwise}. \\ \end{array} \right.$ (C.8)

To calculate a probability, the PDF must be integrated over one or more intervals. As follows from Lebesgue integration theory (``measure theory''), the probability of any countably infinite set of discrete points is zero when the PDF is finite. This is because such a set of points is a ``set of measure zero'' under integration. Note that we write $ \hat{p}(x)$ for discrete probability distributions and $ p(x)$ for PDFs. A discrete probability distribution such as that in (C.4) can be written as

$\displaystyle p(x) = \frac{1}{2}\delta(x) + \frac{1}{2}\delta(x-1)$ (C.9)

where $ \delta(x)$ denotes an impulse.C.1


Stochastic Process

(Again, for a more complete treatment, see [201] or the like.)


Definition: A stochastic process $ x$ is defined as a sequence of random variables $ x(n)$ , $ n=\ldots, -2,-1,0,1,2,\ldots\,$ .

A stochastic process may also be called a random process, noise process, or simply signal (when the context is understood to exclude deterministic components).


Stationary Stochastic Process


Definition: We define a stationary stochastic process $ x(n)$ , $ n=0,\pm1,\pm2,\ldots$ as a stochastic process consisting of identically distributed random variables $ x(n)$ . In particular, all statistical measures are time-invariant.

When a stochastic process is stationary, we may measure statistical features by averaging over time. Examples below include the sample mean and sample variance.


Expected Value


Definition: The expected value of a continuous random variable $ v\in(-\infty,\infty)$ is denoted $ E\{v\}$ and is defined by

$\displaystyle E\{v\} \isdef \int_{-\infty}^\infty x \, p_v(x) dx$ (C.12)

where $ p_v(x)$ denotes the probability density function (PDF) for the random variable v.


Example: Let the random variable $ v(n)$ be uniformly distributed between $ a$ and $ b$ , i.e.,

$\displaystyle p_v(x) = \left\{\begin{array}{ll} \frac{1}{b-a}, & a\leq x \leq b \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array} \right.$ (C.13)

Then the expected value of $ v(n)$ is computed as

$\displaystyle E\{v\} = \int_a^b x \frac{1}{b-a} dx = \frac{1}{2}\frac{b^2-a^2}{b-a} = \frac{b+a}{2}.$ (C.14)

Thus, the expected value of a random variable uniformly distributed between $ a$ and $ b$ is simply the average of $ a$ and $ b$ .

For a stochastic process, which is simply a sequence of random variables, $ E\{x(n)\}$ means the expected value of $ x(n)$ over ``all realizations'' of the random process $ x(\cdot)$ . This is also called an ensemble average. In other words, for each ``roll of the dice,'' we obtain an entire signal $ x(n),\,
n=0,\pm1,\pm2,\cdots$ , and to compute $ E\{x(0)\}$ , say, we average together all of the values of $ x(0)$ obtained for all ``dice rolls.''

For a stationary random process $ x = \{x(n),\,
n=0,\pm1,\pm2,\cdots\}$ , the random variables $ x(n)$ which make it up are identically distributed. As a result, we may normally compute expected values by averaging over time within a single realization of the random process, instead of having to average ``vertically'' at a single time instant over many realizations of the random process.C.2 Denote time averaging by

$\displaystyle {\cal E}_n\{x(n)\} \isdef \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N x(n).$ (C.15)

Then, for a stationary random processes, we have $ E\{x(n)\} =
{\cal E}_n\{x(n)\}$ . That is, for stationary random signals, ensemble averages equal time averages.

We are concerned only with stationary stochastic processes in this book. While the statistics of noise-like signals must be allowed to evolve over time in high quality spectral models, we may require essentially time-invariant statistics within a single frame of data in the time domain. In practice, we choose our spectrum analysis window short enough to impose this. For audio work, 20 ms is a typical choice for a frequency-independent frame length.C.3 In a multiresolution system, in which the frame length can vary across frequency bands, several periods of the band center-frequency is a reasonable choice. As discussed in §5.5.2, the minimum number of periods required under the window for resolution of spectral peaks depends on the window type used.


Mean


Definition: The mean of a stochastic process $ v(n)$ at time $ n$ is defined as the expected value of $ v(n)$ :

$\displaystyle \mu_{v(n)} \isdef E\{v(n)\} \isdef \int_{-\infty}^\infty x p_{v(n)}(x) dx$ (C.16)

where $ p_{v(n)}(x)$ is the probability density function for the random variable $ v(n)$ .

For a stationary stochastic process $ v$ , the mean is given by the expected value of $ v(n)$ for any $ n$ . I.e., $ \mu_v = E\{v(n)\}$ for all $ n$ .


Sample Mean


Definition: The sample mean of a set of $ N$ samples from a particular realization of a stationary stochastic process $ v$ is defined as the average of those samples:

$\displaystyle \hat{\mu}_{v} \isdef {\cal E}_N\{v(0:N-1)\} \isdef \frac{1}{N}\sum_{n=0}^{N-1} v(n)$ (C.17)

For a stationary stochastic process $ v$ , the sample mean is an unbiased estimator of the mean, i.e.,

$\displaystyle E\{\hat{\mu}_{v}\} = \mu_v.$ (C.18)


Variance


Definition: The variance or second central moment of a stochastic process $ v(n)$ at time $ n$ is defined as the expected value of $ \left\vert v(n)-\mu_{v(n)}\right\vert^2$ :

$\displaystyle \sigma^2_{v(n)} \isdef E\{\left\vert v(n)-\mu_{v(n)}\right\vert^2\} \isdef \int_{-\infty}^\infty \left\vert v(n)-\mu_{v(n)}\right\vert^2 p_{v(n)}(x) dx$ (C.19)

where $ p_{v(n)}(x)$ is the probability density function for the random variable $ v(n)$ .

For a stationary stochastic process $ v$ , the variance is given by the expected value of $ \left\vert v(n)-\mu_v\right\vert^2$ for any $ n$ .


Sample Variance


Definition: The sample variance of a set of $ N$ samples from a particular realization of a stationary stochastic process $ v$ is defined as average squared magnitude after removing the known mean:

$\displaystyle \hat{\sigma}^2_{v} \isdef {\cal E}_N\{\left\vert v(n)-\mu_v\right\vert^2\} \isdef \frac{1}{N}\sum_{n=0}^{N-1} \left\vert v(n)-\mu_v\right\vert^2 = \frac{1}{N}\sum_{n=0}^{N-1} \left\vert v(n)\right\vert^2 -\mu_v^2$ (C.20)

The sample variance is a unbiased estimator of the true variance when the mean is known, i.e.,

$\displaystyle E\{\hat{\sigma}^2_{v}\} = \sigma^2_v.$ (C.21)

This is easy to show by taking the expected value:
$\displaystyle E\{\hat{\sigma}^2_{v}\}$ $\displaystyle =$ $\displaystyle E{\cal E}_N\{\left\vert v(n)-\mu_v\right\vert^2\} = {\cal E}_N\{E\left\vert v(n)-\mu_v\right\vert^2\}$  
  $\displaystyle =$ $\displaystyle {\cal E}_N\{E\left\vert v(n)\right\vert^2-E\overline{v(n)}\mu_v-Ev(n)\overline{\mu_v}+\left\vert\mu_v\right\vert^2\}$  
  $\displaystyle =$ $\displaystyle {\cal E}_N\{\sigma_v^2+\left\vert\mu_v\right\vert^2-\overline{\mu_v}\mu_v-\mu_v\overline{\mu_v}+\left\vert\mu_v\right\vert^2\}$  
  $\displaystyle =$ $\displaystyle {\cal E}_N\{\sigma_v^2\} = \sigma^2_v.
\protect$ (C.22)

When the mean is unknown, the sample mean is used in its place:

$\displaystyle \hat{\sigma}^2_{v} \isdef \frac{1}{N-1}\sum_{n=0}^{N-1} \left\vert v(n)-\hat{\mu}_v\right\vert^2$ (C.23)

The normalization by $ N-1$ instead of $ N$ is necessary to make the sample variance be an unbiased estimator of the true variance. This adjustment is necessary because the sample mean is correlated with the term $ v(n)$ in the sample variance expression. This is revealed by replacing $ \mu_v$ with $ \hat{\mu}_v$ in the calculation of (C.22).


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Correlation Analysis
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Relation of Smoothness to Roll-Off Rate