Coupon Accepted Successfully!


Random Signal And Random Process


Cumulative Distribution Function (CDF)


Properties of CDF :

  1. 0 ≤ Fx (x) ≤ 1.
  2. Fx (x) is non-decreasing function in nature.
  3. Fx (–∞) = 0 and Fx (∞) = 1
  4. P(a < X ≤ b) = Fx (b) – Fx (a)

Probability Distribution Function (PDF)


Properties of PDF :

  1. fx (x) ≥ 0.
  2. 534.png 
  3. 539.png 

Joint Cumulative Distribution Function (JCDF)

Fxy (x, y) = P (X ≤ x, Y ≤ y)

Properties :

  1. Fxy (x, y) ≥ 0.
  2. The Joint Cumulative Distribution Function is a monotonous non-decreasing function of x and y.
  3. It is always continous everywhere in xy-plane.

Joint Probability Distribution Function (JPDF)

pxy (x, y) = 544.png

Properties of Joint PDF

  1. pxy (x, y) ≥ 0.
  2. 550.png 
  3. The joint PDF is continuous everywhere because joint CDF is continuous.

Relationship between Joint PDF’s and Probability

  1. P(x1 < X ≤ x2) = 555.png
  2. P(x1 < X ≤ x2, y1 < Y ≤ y2) = 560.png

Marginal Densities

Fx(x) = P(X ≤ x) = P(X ≤ x, –∞ < y < ∞)



Fy(y) = P(Y ≤ y) = P(Y ≤ y, –∞ < x < ∞)


Conditional Probabilities




Properties of Conditional Probability :

  1. 585.png ≥ 0 and 590.png ≥ 0.
  2. 595.png and 601.png
  3. If random variable X and Y are statistically independent, then
    606.png = py(y) and 611.png = px(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.

px(x) = 616.png


Gaussian or Normal Random Variable

This is a continuous random variable whose pdf is given as below:–

px(x) = 626.png


Gaussian Random Variable

Properties of Gaussian Random Variable :

  1. The peak value occurs at x = m
    px(m) = 636.png where m is mean value.
  2. Gaussian pdf shows even symmetry, i.e.
    px (m – k) = px (m + k).
  3. The area under pdf of Gaussian random variable is 641.png for all value of x below m and 646.png for all value of x above m, i.e.P(X ≤ m) = P(X > m) = 652.png

Rayleigh Distribution


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

Its pdf is as follows:

pz(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:

Rt (t1, t2) = E[X(v1 t1) . (v, t2)]

Properties of Autocorrelation Function

  1. Rx (t) = Rx (–t), i.e. it is an even function.
  2. Maximum absolute value is achieved at ô = 0, i.e. |Rx (ô)| ≤ Rx (0).
  3. If for any time t1, we have Rx (t1) = Rx (0), then for all integers k, we have Rx (kT0) = Rx (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:

Rxy (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:

Sx (w) = 667.png

Cross power spectral density is given by

Sxy (w) = 672.png

Rxy (t) is cross-correlation function of random process X and Y.

Properties of Power Spectral Density :

  1. Sx (w) is a real function of w.
  2. Power spectral density of random process X(t) is even function of frequency, i.e. Sx (w) = Sx (–w)
  3. Power spectral density of a random process X(t) is a non-negative function of w.

Sx (w) ≥ 0 for all w.


Information Capacity Theorem


where, P = average transimitted power

Test Your Skills Now!
Take a Quiz now
Reviewer Name