# Random Signal And Random Process

**Cumulative Distribution Function (CDF)**

Properties of CDF :

- 0 â‰¤ F
_{x}(x) â‰¤ 1. - F
_{x}(x) is non-decreasing function in nature. - F
_{x}(â€“âˆž) = 0 and F_{x}(âˆž) = 1 - P(a < X â‰¤ b) = F
_{x}(b) â€“ F_{x}(a)

**Probability Distribution Function (PDF)**

Properties of PDF :

- f
_{x}(x) â‰¥ 0.

**Joint Cumulative Distribution Function (JCDF)**

F_{xy} (x, y) = P (X â‰¤ x, Y â‰¤ y)

Properties :

- F
_{xy}(x, y) â‰¥ 0. - The Joint Cumulative Distribution Function is a monotonous non-decreasing function of x and y.
- It is always continous everywhere in xy-plane.

**Joint Probability Distribution Function (JPDF)**

p_{xy} (x, y) =

Properties of Joint PDF

- p
_{xy}(x, y) â‰¥ 0. - The joint PDF is continuous everywhere because joint CDF is continuous.

# Relationship between Joint PDFâ€™s and Probability

- P(x
_{1}< X â‰¤ x_{2}) = - P(x
_{1}< X â‰¤ x_{2}, y_{1}< Y â‰¤ y_{2}) =

Marginal Densities

F_{x}(x) = P(X â‰¤ x) = P(X â‰¤ x, â€“âˆž < y < âˆž)

=

Similarly,

F_{y}(y) = P(Y â‰¤ y) = P(Y â‰¤ y, â€“âˆž < x < âˆž)

=

Conditional Probabilities

Similarly,

Properties of Conditional Probability :

- â‰¥ 0 and â‰¥ 0.
- and
- If random variable X and Y are statistically independent, then
= p
_{y}(y) and = p_{x}(x)

# Important Random Variables

**Uniform Random Variable**

This is a continuous random variable whose probability distribution function is constant between two values a and b and zero elsewhere.

p_{x}(x) =

Gaussian or Normal Random Variable

This is a continuous random variable whose pdf is given as below:â€“

p_{x}(x) =

Gaussian Random Variable

Properties of Gaussian Random Variable :

- The peak value occurs at x = m

p_{x}(m) = where m is mean value. - Gaussian pdf shows even symmetry, i.e.
p
_{x}(m â€“ k) = p_{x}(m + k). - The area under pdf of Gaussian random variable is for all value of x below m and for all value of x above m, i.e.P(X â‰¤ m) = P(X > m) =

# Rayleigh Distribution

Rayleigh distribution is a continuous random variable that is produced from two random variable X and Y.

Its pdf is as follows:

p_{z}(z) =

Rayleigh Distribution

# Autocorrelation Function

This function as the name suggest provide the relation of a function with its shifted version.

**Mathematically, it may be expressed as:**

R_{t} (t_{1}, t_{2}) = E[X(v_{1} t_{1}) . (v, t_{2})]

Properties of Autocorrelation Function

- R
_{x}(t) = R_{x}(â€“t), i.e. it is an even function. - Maximum absolute value is achieved at Ã´ = 0, i.e. |R
_{x}(Ã´)| â‰¤ R_{x}(0). - If for any time t
_{1}, we have R_{x}(t_{1}) = R_{x}(0), then for all integers k, we have R_{x}(kT_{0}) = R_{x}(0).

**Cross-Correlation Function**

An autocorrelation is defined for a single random process whereas cross-correlation is defined for two random processes. Let us consider two random process X(t) and Y(t), then cross-correlation between X(t) and Y(t) may be defined as:

R_{xy} (t, u) = E [X(t) Y(u)]

# Spectral Densities and Power Spectral Density

Spectral Density is used to represent random process in frequency domain.

Power Spectral Density (PSD)

Power Spectral Density of random process X(t) is given as:

S_{x} (w) =

Cross power spectral density is given by

S_{xy} (w) =

R_{xy} (t) is cross-correlation function of random process X and Y.

Properties of Power Spectral Density :

- S
_{x}(w) is a real function of w. - Power spectral density of random process X(t) is even function of frequency, i.e. S
_{x}(w) = S_{x}(â€“w) - Power spectral density of a random process X(t) is a non-negative function of w.

S_{x} (w) â‰¥ 0 for all w.

Information Capacity Theorem

* *

where, P = average transimitted power