Probability Theory and Stochastic processing (EC-703 (C)) - Unit 5 Short Notes

UNIT 5: PROBABILITY THEORY & STOCHASTIC PROCESSES

Based on EC-703(C) - Nov 2023 Exam Analysis

I. FOUNDATIONS OF PROBABILITY & RANDOM EVENTS

Axioms of Probability

For a sample space $S$ and event $A \subseteq S$, probability $P(A)$ satisfies:

1. Non-negativity: $P(A) \geq 0$

2. Normalization: $$\displaystyle P(S) = 1 $$

3. Additivity: For mutually exclusive events $$\displaystyle A_i $$, $$\displaystyle P\left(\bigcup_i A_i\right) = \sum_i P(A_i) $$

Conditional Probability & Bayes' Theorem

Definition: $$\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)} $$, provided $$\displaystyle P(B) > 0 $$.

Bayes' Theorem:

If $$\displaystyle B_1, B_2, \dots, B_n $$ form a partition of $S$, then

$$ P(B_i|A) = \frac{P(A|B_i)P(B_i)}{\sum_{j=1}^n P(A|B_j)P(B_j)} $$

Proof: From definition,

$$ P(B_i|A) = \frac{P(A \cap B_i)}{P(A)} = \frac{P(A|B_i)P(B_i)}{P(A)}. $$

Using Total Probability Theorem:

$$ P(A) = \sum_{j=1}^n P(A|B_j)P(B_j). $$

Substitute to get Bayes' formula. \boxed{P(B_i|A) = \frac{P(A|B_i)P(B_i)}{\sum_{j} P(A|B_j)P(B_j)}}

Exam Tip (Q1.a): State theorem clearly, then prove using definitions of conditional probability and total probability.

Statistical Independence of Events

Events $A$ and $B$ are independent iff:

$$ P(A \cap B) = P(A)P(B) $$

Properties (Q1.b):

If $A$ and $B$ are independent, then:

1. $A'$ and $B$ are independent.

2. $A$ and $B'$ are independent.

3. $A'$ and $B'$ are independent.

Proof Sketch for (1):

$$ P(A' \cap B) = P(B) - P(A \cap B) = P(B) - P(A)P(B) = P(B)(1-P(A)) = P(A')P(B). $$

Similarly for others.

Application (Q2.a: Missile Launch):

Let $A$: "Relay A fails", $B$: "Relay B fails".

Given: $$\displaystyle P(A)=0.01 $$, $$\displaystyle P(B)=0.03 $$, $$\displaystyle P(B|A)=0.06 $$.

• Accidental launch: $$\displaystyle P(A \cap B) = P(A)P(B|A) = 0.01 \times 0.06 = 0.0006 $$

• $P(A|B)$: $$\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)} $$. First find $P(A \cap B)$ as above, then $$\displaystyle P(A|B) = 0.0006 / 0.03 = 0.02 $$.

• Independence? Check $$\displaystyle P(A \cap B) \stackrel{?}{=} P(A)P(B) $$. $$\displaystyle P(A)P(B)=0.01 \times 0.03 = 0.0003 \neq 0.0006 $$. Hence not independent.

II. RANDOM VARIABLES (RVs) & PROBABILITY DISTRIBUTIONS

Classification of RVs

• Discrete RV: Takes countable values (e.g., Poisson, Binomial).

• Continuous RV: Takes values in intervals (e.g., Gaussian, Uniform).

PDF & CDF

For continuous RV $X$:

• PDF: $$\displaystyle f_X(x) \geq 0 $$, $$\displaystyle \int_{-\infty}^{\infty} f_X(x)dx = 1 $$.

• CDF: $$\displaystyle F_X(x) = P(X \leq x) = \int_{-\infty}^{x} f_X(t)dt $$.

• Relationship: $$\displaystyle f_X(x) = \frac{d}{dx}F_X(x) $$ (where differentiable).

Important Distributions

1. Uniform Distribution (Q4.a context)

$X \sim \mathcal{U}(a,b)$:

$$ f_X(x) = \begin{cases} \frac{1}{b-a} & a \leq x \leq b \\ 0 & \text{otherwise} \end{cases} $$

For $$\displaystyle a=-1, b=1 $$: $$\displaystyle f_X(x) = \frac{1}{2}, \, -1 \leq x \leq 1 $$.

2. Gaussian (Normal) Distribution (Q2.b)

$$\displaystyle X \sim \mathcal{N}(\mu, \sigma^2) $$:

$$ f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

Standardization: $$\displaystyle Z = \frac{X-\mu}{\sigma} \sim \mathcal{N}(0,1) $$. Probability Calculation: $$\displaystyle P(a < X < b) = P\left(\frac{a-\mu}{\sigma} < Z < \frac{b-\mu}{\sigma}\right) = \Phi\left(\frac{b-\mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right) $$, where $\Phi$ is standard normal CDF.

Example (Q2.b): $$\displaystyle \mu=1, \sigma^2=4 \Rightarrow \sigma=2 $$. Find $$\displaystyle P(1<X<2) $$.

$$\displaystyle Z_1 = \frac{1-1}{2}=0 $$, $$\displaystyle Z_2 = \frac{2-1}{2}=0.5 $$.

$$\displaystyle P(1<X<2) = \Phi(0.5) - \Phi(0) = 0.6915 - 0.5 = 0.1915 $$.

3. Rayleigh Distribution (Q3.a)

$$\displaystyle f_X(x) = \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}, \, x \geq 0 $$. Validation as PDF:

1. Non-negative: clear for $x \geq 0$.

2. Integral = 1:

$$ \int_0^\infty \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)} dx. $$

Let $$\displaystyle u = x^2/(2\sigma^2) \Rightarrow du = \frac{x}{\sigma^2}dx $$.

Integral becomes $$\displaystyle \int_0^\infty e^{-u} du = 1 $$. ✓

4. Poisson Distribution (Q3.b context)

PMF: $$\displaystyle P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}, \, k=0,1,2,\dots $$ $\lambda$: average rate.

Moments & Moment Generating Function (MGF)

Definition (Q4.b): MGF of RV $X$ is $$\displaystyle M_X(t) = E[e^{tX}] $$, for $t$ in neighborhood of 0.

Properties (State & Prove):

1. Uniqueness: MGF uniquely determines distribution (if exists in interval around 0).

2. Moments: $$\displaystyle E[X^n] = M_X^{(n)}(0) $$ (n-th derivative at 0).

3. Sum of Independent RVs: If $X,Y$ independent, $$\displaystyle M_{X+Y}(t) = M_X(t) M_Y(t) $$.

MGF of Poisson (Q3.b):

$$ M_X(t) = \sum_{k=0}^\infty e^{tk} \frac{\lambda^k e^{-\lambda}}{k!} = e^{-\lambda} \sum_{k=0}^\infty \frac{(\lambda e^t)^k}{k!} = e^{-\lambda} e^{\lambda e^t} = e^{\lambda(e^t - 1)}. $$

Functions of a Single RV (Transformation)

For $$\displaystyle Z = g(X) $$:

• If $g$ monotonic: $$\displaystyle f_Z(z) = f_X(x) \left| \frac{dx}{dz} \right| $$, where $$\displaystyle x = g^{-1}(z) $$.

• General: $$\displaystyle F_Z(z) = P(g(X) \leq z) = \int_{\{x: g(x) \leq z\}} f_X(x)dx $$.

III. JOINT DISTRIBUTION & TRANSFORMATIONS

Joint Probability Distributions

For two RVs $(X,Y)$:

• Joint CDF: $$\displaystyle F_{X,Y}(x,y) = P(X \leq x, Y \leq y) $$.

• Joint PDF (continuous): $$\displaystyle f_{X,Y}(x,y) = \frac{\partial^2}{\partial x \partial y} F_{X,Y}(x,y) $$.

Properties of Joint CDF (Q5.a):

1. $$\displaystyle F_{X,Y}(x,y) $$ is non-decreasing in $x$ and $y$.

2. $$\displaystyle F_{X,Y}(-\infty, y) = 0 $$, $$\displaystyle F_{X,Y}(x, -\infty) = 0 $$, $$\displaystyle F_{X,Y}(\infty, \infty)=1 $$.

3. For $$\displaystyle x_1 < x_2, y_1 < y_2 $$:

$$ P(x_1 < X \leq x_2, y_1 < Y \leq y_2) = F_{X,Y}(x_2,y_2) - F_{X,Y}(x_1,y_2) - F_{X,Y}(x_2,y_1) + F_{X,Y}(x_1,y_1) \geq 0. $$

Independence of Random Variables

$X$ and $Y$ independent iff:

• $$\displaystyle F_{X,Y}(x,y) = F_X(x)F_Y(y) $$ for all $x,y$,

or (for continuous with PDFs):

• $$\displaystyle f_{X,Y}(x,y) = f_X(x) f_Y(y) $$ for all $x,y$.

Functions of Two RVs (Sum: Convolution)

For $$\displaystyle Z = X+Y $$, with $X,Y$ independent:

$$ f_Z(z) = \int_{-\infty}^{\infty} f_X(x) f_Y(z-x) dx \quad \text{(convolution)}. $$

Example (Q4.a): $X,Y \sim \mathcal{U}(-1,1)$ independent. $$\displaystyle f_X(x) = f_Y(y) = \frac{1}{2} $$ for $|x|,|y| \leq 1$. $$\displaystyle f_Z(z) = \int \frac{1}{2} \cdot \frac{1}{2} \, dx $$ over $x$ such that $|x|\leq 1$ and $|z-x|\leq 1$.

Integration limits depend on $z$:

• For $$\displaystyle |z| < 2 $$: triangular PDF.

• Result:

$$ f_Z(z) = \begin{cases} \frac{2-|z|}{4} & |z| \leq 2 \\ 0 & \text{otherwise} \end{cases} $$

Conditional Distributions & Expectations

• Conditional PDF: $$\displaystyle f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)} $$ (if $$\displaystyle f_X(x)>0 $$).

• Conditional Expectation: $$\displaystyle E[Y|X=x] = \int y f_{Y|X}(y|x) dy $$.

IV. STOCHASTIC PROCESSES (RANDOM PROCESSES)

Definition & Classification

A random process is a collection of RVs $\{X(t), t \in T\}$ indexed by time $T$.

• Classification:

  • By time: Continuous-time ($T$ continuous), Discrete-time ($T$ discrete).

  • By amplitude: Continuous-valued, Discrete-valued.

  • Deterministic: Evolves predictably (e.g., $$\displaystyle X(t)=A\cos\omega t $$ with fixed $A,\omega$).

  • Non-deterministic: Contains randomness (e.g., $$\displaystyle X(t)=A\cos\omega t $$ with random $A$).

Stationarity

1. Strict-Sense Stationarity (SSS)

Definition: For all $n$, all $$\displaystyle t_1,\dots,t_n $$, and all $\tau$,

$$ F_{X(t_1),\dots,X(t_n)}(x_1,\dots,x_n) = F_{X(t_1+\tau),\dots,X(t_n+\tau)}(x_1,\dots,x_n). $$

Verification (Q6.a): $$\displaystyle X(t)=A\cos\omega t $$, $A \sim \mathcal{U}(0,\pi)$.

• Joint distribution of any finite set depends on $$\displaystyle t_i $$ through $$\displaystyle \cos\omega t_i $$, but since $A$ is fixed (non-random amplitude), the distribution changes with $t$ (e.g., mean $$\displaystyle E[X(t)] = E[A]\cos\omega t \neq \text{constant} $$). Hence not SSS (and not WSS either, as shown below).

2. Wide-Sense Stationarity (WSS)

Definition:

1. Constant mean: $$\displaystyle E[X(t)] = \mu_X $$ (independent of $t$).

2. Autocorrelation depends only on $$\displaystyle \tau = t_2-t_1 $$:

$$ R_{XX}(t_1,t_2) = R_{XX}(\tau). $$

Example (Q6.b): $$\displaystyle X(t)=\cos(\omega t + \theta) $$, $\theta \sim \mathcal{U}(0, \pi/2)$.

• Mean: $$\displaystyle E[X(t)] = E[\cos(\omega t + \theta)] = \cos(\omega t)E[\cos\theta] - \sin(\omega t)E[\sin\theta] $$.

  $$\displaystyle E[\cos\theta] = \frac{2}{\pi} $$, $$\displaystyle E[\sin\theta] = \frac{2}{\pi} $$? Actually compute:

  $$\displaystyle E[\cos\theta] = \int_0^{\pi/2} \cos\theta \cdot \frac{2}{\pi} d\theta = \frac{2}{\pi} [\sin\theta]_0^{\pi/2} = \frac{2}{\pi} $$.

  $$\displaystyle E[\sin\theta] = \frac{2}{\pi} [-\cos\theta]_0^{\pi/2} = \frac{2}{\pi} $$.

  So $$\displaystyle E[X(t)] = \frac{2}{\pi} (\cos\omega t - \sin\omega t) $$, which depends on $t$. Hence not WSS.

• Average Power: $$\displaystyle P = E[X^2(t)] = E[\cos^2(\omega t+\theta)] = E\left[\frac{1+\cos(2\omega t+2\theta)}{2}\right] = \frac{1}{2} + \frac{1}{2}E[\cos(2\omega t+2\theta)] $$.

  $$\displaystyle E[\cos(2\omega t+2\theta)] = \cos(2\omega t)E[\cos2\theta] - \sin(2\omega t)E[\sin2\theta] $$.

  Compute $$\displaystyle E[\cos2\theta] = \frac{2}{\pi}\int_0^{\pi/2} \cos2\theta d\theta = \frac{2}{\pi}[\frac{\sin2\theta}{2}]_0^{\pi/2}=0 $$.

  Similarly $$\displaystyle E[\sin2\theta]=0 $$. So $$\displaystyle E[X^2(t)] = \frac{1}{2} $$.

  \boxed{\text{Average Power} = \frac{1}{2}}.

Autocorrelation Function (ACF)

Definition: $$\displaystyle R_{XX}(t_1,t_2) = E[X(t_1)X(t_2)] $$.

For WSS: $$\displaystyle R_{XX}(\tau) = E[X(t)X(t+\tau)] $$.

Properties:

1. $$\displaystyle R_{XX}(\tau) $$ is even: $$\displaystyle R_{XX}(-\tau) = R_{XX}(\tau) $$.

2. Maximum at $$\displaystyle \tau=0 $$: $$\displaystyle |R_{XX}(\tau)| \leq R_{XX}(0) $$ (by Cauchy-Schwarz).

3. Positive semi-definite: For any $n$, $$\displaystyle \tau_1,\dots,\tau_n $$, and complex $$\displaystyle c_i $$, $$\displaystyle \sum_{i,j} c_i \overline{c_j} R_{XX}(\tau_i-\tau_j) \geq 0 $$.

Ergodicity (Q5.b)

• Time Average: $$\displaystyle \overline{X} = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T} X(t) dt $$.

• Ensemble Average: $E[X(t)]$ (mean function).

• Ergodic in mean: Time average equals ensemble average (for all sample functions, with probability 1).

Concept: Ergodicity allows replacing ensemble averages with time averages from a single observation.

V. SPECTRAL ANALYSIS OF RANDOM PROCESSES

Power Spectral Density (PSD) (Q7.a)

Definition: $$\displaystyle S_{XX}(\omega) = \mathcal{F}\{R_{XX}(\tau)\} = \int_{-\infty}^{\infty} R_{XX}(\tau) e^{-j\omega\tau} d\tau $$. Wiener-Khinchin Theorem: For WSS process, $$\displaystyle R_{XX}(\tau) $$ and $$\displaystyle S_{XX}(\omega) $$ form a Fourier transform pair. Proof Sketch: Start from $$\displaystyle R_{XX}(\tau) = E[X(t)X(t+\tau)] $$. Write $X(t)$ as inverse Fourier transform of its spectrum (if it exists), then compute expectation and use Fubini's theorem to swap integrals. Result:

$$ R_{XX}(\tau) = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{XX}(\omega) e^{j\omega\tau} d\omega. $$

Properties of PSD:

1. $$\displaystyle S_{XX}(\omega) \geq 0 $$ (non-negative).

2. $$\displaystyle S_{XX}(\omega) $$ is real and even: $$\displaystyle S_{XX}(-\omega) = S_{XX}^*(\omega) = S_{XX}(\omega) $$.

3. Total Power: $$\displaystyle R_{XX}(0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{XX}(\omega) d\omega $$.

Cross-Correlation & Cross-Power Spectral Density (Q8.a)

• Cross-correlation: $$\displaystyle R_{XY}(\tau) = E[X(t)Y(t+\tau)] $$.

• Cross-PSD: $$\displaystyle S_{XY}(\omega) = \mathcal{F}\{R_{XY}(\tau)\} $$.

Properties of Cross-Power Spectrum:

1. Conjugate symmetry: $$\displaystyle S_{XY}(\omega) = S_{YX}^*(-\omega) $$.

2. Not necessarily even or real.

3. Cauchy-Schwarz: $$\displaystyle |S_{XY}(\omega)|^2 \leq S_{XX}(\omega) S_{YY}(\omega) $$.

4. If $X$ and $Y$ are orthogonal ($$\displaystyle R_{XY}(\tau)=0 $$), then $$\displaystyle S_{XY}(\omega)=0 $$.

Linear System Interaction (Q7.b)

Setup: Input $X(t)$ → LTI system with frequency response $H(\omega)$ → Output $Y(t)$.

Then $$\displaystyle Y(t) = X(t) * h(t) $$, where $h(t)$ is impulse response.

Proof of $$\displaystyle S_{YY}(\omega) = |H(\omega)|^2 S_{XX}(\omega) $$:

1. $$\displaystyle R_{YY}(\tau) = E[Y(t)Y(t+\tau)] = E[(X*h)(t) \cdot (X*h)(t+\tau)] $$.

2. Using convolution and expectation:

$$ R_{YY}(\tau) = \iint h(\alpha) h(\beta) R_{XX}(\tau + \beta - \alpha) d\alpha d\beta. $$

3. Take Fourier transform:

$$ S_{YY}(\omega) = H(\omega) H^*(\omega) S_{XX}(\omega) = |H(\omega)|^2 S_{XX}(\omega). $$

Input-Output Cross-Spectrum:

$$ S_{XY}(\omega) = H(\omega) S_{XX}(\omega). $$

VI. ADVANCED TOPICS & APPLICATIONS (From Q5-Q8)

Classification of Random Processes (Q8.b)

1. By stationarity:

   • SSS: All finite-dimensional distributions invariant to time shift.

   • WSS: Constant mean, autocorrelation depends only on $\tau$.

   • Non-stationary: Mean and/or autocorrelation vary with absolute time.

2. By ergodicity:

   • Ergodic: Time averages equal ensemble averages (e.g., ergodic in mean, autocorrelation).

   • Non-ergodic: Not all time averages converge to ensemble averages (e.g., $$\displaystyle X(t)=A $$ with random constant $A$).

3. By continuity:

   • Continuous-time: $t$ continuous.

   • Discrete-time: $t$ discrete (e.g., $X[n]$).

   • Mixed: Combination (e.g., jump processes).

4. By amplitude:

   • Continuous: $X(t)$ continuous-valued (e.g., Gaussian process).

   • Discrete: $X(t)$ takes discrete values (e.g., Poisson process).

   • Mixed: Both types present.

Practical Interpretation

• Stationarity: Many signal processing tools (e.g., PSD) assume WSS. Non-stationary signals require time-frequency analysis (e.g., spectrogram).

• Ergodicity: Enables analysis of a single, long observation instead of multiple realizations. Crucial in communication systems for estimating channel statistics.

• PSD in Communications: Measures power distribution over frequency. Used for:

  • Bandwidth determination: Find frequency band containing most power.

  • Noise analysis: White noise has flat PSD; colored noise has shaped PSD.

  • Signal design: Minimize interference by shaping PSD.

[!TIP] Common Pitfalls:

• Independence vs. Uncorrelated: Independent ⇒ uncorrelated, but converse not true (except for Gaussian RVs).

• WSS vs. SSS: WSS is weaker; all SSS processes are WSS, but not vice versa.

• Ergodicity: A process can be WSS but not ergodic (e.g., $$\displaystyle X(t)=A $$ with random $A$: mean is constant $E[A]$, but time average of a sample function is $A$ itself, not $E[A]$).

• PSD Properties: $$\displaystyle S_{XX}(\omega) \geq 0 $$ always; $$\displaystyle S_{XX}(\omega) $$ real & even only if $$\displaystyle R_{XX}(\tau) $$ is real & even (which it is for real processes).

• Convolution for Sum: For two independent continuous RVs, $$\displaystyle f_{X+Y}(z) = \int f_X(x)f_Y(z-x)dx $$. Always check integration limits based on supports.