Stochastic Processes

Abstract

This chapter is an extension of the previous chapter. In the previous chapter, we focused essentially on random variables. In this chapter, we introduce the concept of random (or stochastic) process as a generalization of a random variable to include another dimension—time. While a random variable depends on the outcome of a random experiment, a random process depends on both the outcome of a random experiment and time. In other words, if a random variable X is time-dependent, X(t) is known as a random process. Thus, a random process may be regarded as any process that changes with time and controlled by some probabilistic law. For example, the number of customers N in a queueing system varies with time; hence N(t) is a random process

For me problem-solving is the most interesting thing in life. To be handed something that’s a complete mess and straighten it out. To organize where there is no organization. To give form to a medium that has no form.

—Sylvester Weaver

Problems

• 3.1 If X(t) = A sin4t, where A is random variable uniformly distributed between 0 and 2, find E[X(t)] and E[X²(t)].

• 3.2 Given a random process X(t) = At + 2, where A is a random variable uniformly distributed over the range (0,1),

  • (a) sketch three sample functions of X(t),

  • (b) find $\overline{X(t)}\kern0.75em \mathrm{and}\kern0.5em \overline{X^2(t)}$,

  • (c) determine R _X (t ₁,t ₂),

  • (d) Is X(t) WSS?

• 3.3 If a random process is given by

  $$ X(t)=A \cos \omega t-B \sin \omega t, $$

  where ω is a constant and A and B are independent Gaussian random variables with zero mean and variance σ², determine: (a) E[X], E[X²] and Var(X), (b) the autocorrelation function R _X (t ₁,t ₂).

• 3.4 Let Y(t) = X(t − 1) + cos3t, where X(t) is a stationary random process. Determine the autocorrelation function of Y(t) in terms of R _X (τ).

• 3.5 If Y(t) = X(t) − X(t − α), where α is a constant and X(t) is a random process. Show that

  $$ \kern1em {R}_Y\left({t}_1,{t}_2\right)={R}_x\left({t}_1,{t}_2\right)-{R}_x\left({t}_1,{t}_2-\alpha \right)-{R}_x\left({t}_1-\alpha, {t}_2\right)+{R}_x\left({t}_1-\alpha, {t}_2-\alpha \right) $$

• 3.6 A random stationary process X(t) has mean 4 and autocorrelation functon

  $$ {R}_X\left(\tau \right)=5{e}^{-2\left|\tau \right|} $$

  • (a) If Y(t) = X(t − 1), find the mean and autocorrelation function of Y(t).

  • (b) Repeat part (a) if Y(t) = tX(t).

• 3.7 Let Z(t) = X(t) + Y(t), where X(t) and Y(t) are two independent stationary random processes. Find R _Z (τ) in terms of R _X (τ) and R _Y (τ).

• 3.8 Repeat the previous problem if Z(t) = 3X(t) + 4Y(t).

• 3.9 If X(t) = Acosωt, where ω is a constant and A random variables with mean μ and variance σ², (a) find < x(t) > and m_X(t). (b) Is X(t) ergodic?

• 3.10 A random process is defined by

  $$ X(t)=A \cos \omega t-B \sin \omega t, $$

  where ω is a constant and A and B are independent random variable with zero mean. Show that X(t) is stationary and also ergodic.

• 3.11 N(t) is a stationary noise process with zero mean and autocorrelation function

  $$ {R}_N\left(\tau \right)=\frac{N_o}{2}\delta \left(\tau \right) $$

  where N_o is a constant. Is N(t) ergodic?

• 3.12 X(t) is a stationary Gaussian process with zero mean and autocorrelation function

  $$ {R}_X\left(\tau \right)={\sigma}^2{e}^{-\alpha \left|\tau \right|} \cos \omega \tau $$

  where σ, ω, and α are constants. Show that X(t) is ergodic.

• 3.13 If X(t) and Y(t) are two random processes that are jointly stationary so that R _XY (t ₁,t ₂) = R _XY (τ), prove that

  $$ {R}_{XY}\left(\tau \right)={R}_{YX}\left(-\tau \right) $$

  where τ = ∣t ₂ − t ₁∣.

• 3.14 For two stationary processes X(t) and Y(t), show that

  • (a)
    $$ \left|{R}_{XY}\left(\tau \right)\right|\le \frac{1}{2}\left[{R}_X(0)+{R}_Y(0)\right] $$
  • (b)
    $$ \left|{R}_{XY}\left(\tau \right)\right|\le \sqrt{R_X(0){R}_Y(0)} $$

• 3.15 Let X(t) and Y(t) be two random processes given by

  X(t) = cos (ωt+Θ)

  Y(t) = sin (ωt+Θ)

  where ω is a constant and Θ is a random variable uniformly distributed over (0,2π). Find

  $$ {R}_{XY}\left(t,t+\tau \right)\kern0.5em \mathrm{and}\kern0.75em {R}_{YX}\left(t,t+\tau \right). $$

• 3.16 X(t) and Y(t) are two random processes described as

  X(t) = A cos ωt + B sin ωt

  Y(t) = B cos ωt − A sin ωt

  where ω is a constant and A = N(0,σ²) and B = N(0,σ²). Find R _XY (τ).

• 3.17 Let X(t) be a stationary random process and Y(t) = X(t) − X(t − a), where a is a constant. Find R _XY (τ).

• 3.18 Let $\left\{\mathit{\mathsf{N}}\left(\mathit{\mathsf{t}}\right),\mathit{\mathsf{t}}\ge 0\right\}$ be a Poisson process with rate λ. Find $\mathit{\mathsf{E}}\left[\mathit{\mathsf{N}}\left(\mathit{\mathsf{t}}\right)\cdotp \mathit{\mathsf{N}}\left(\mathit{\mathsf{t}}+\mathit{\mathsf{s}}\right)\right]$.

• 3.19 For a Poisson process, show that if s < t,

  $$ \mathsf{Prob}\left[\mathsf{N}\left(\mathsf{s}\right)=\mathsf{k}\Big|\mathsf{N}\left(\mathsf{t}\right)=\mathsf{n}\right]=\left(\begin{array}{c}\hfill \mathit{\mathsf{n}}\hfill \\ {}\hfill \mathit{\mathsf{k}}\hfill \end{array}\right){\left(\frac{\mathit{\mathsf{s}}}{\mathit{\mathsf{t}}}\right)}^{\mathit{\mathsf{k}}}{\left(1-\frac{\mathit{\mathsf{s}}}{\mathit{\mathsf{t}}}\right)}^{\mathit{\mathsf{n}}-\mathit{\mathsf{k}}},\mathit{\mathsf{k}}=0,1,\cdots, \mathit{\mathsf{n}} $$

• 3.20 Let N(t) be a renewal process where renewal epochs are Erlang with parameters (m,λ). Show that

  $$ \mathsf{Prob}\left[\mathit{\mathsf{N}}\left(\mathit{\mathsf{t}}\right)=\mathit{\mathsf{n}}\right]={\displaystyle \sum_{\mathit{\mathsf{k}}=\mathit{\mathsf{n}\mathsf{m}}}^{\mathit{\mathsf{n}\mathsf{m}}+\mathit{\mathsf{m}}-1}\frac{{\left(\lambda \mathit{\mathsf{t}}\right)}^{\mathit{\mathsf{k}}}}{\mathit{\mathsf{k}}!}}{\mathit{\mathsf{e}}}^{-\lambda \mathit{\mathsf{t}}} $$

• 3.21 Use MATLAB to generate a random process X(t) = A cos(2πt), where A is a Gaussian random variable with mean zero and variance one. Take 0 < t < 4 s.

• 3.22 Repeat the previous problem if A is random variable uniformly distributed over (−2, 2).

• 3.23 Given that the autocorrelation function ${R}_X\left(\tau \right)=2+3{e}^{-{\tau}^2}$, use MATLAB to plot the function for −2 < τ < 2.

• 3.24 Use MATLAB to generate a random process

  $$ X(t)=2 \cos \left(2\pi t+B\left[n\right]\frac{\pi }{4}\right) $$

  where B[n] is a Bernoulli random sequence taking the values of +1 and −1. Take 0 < t < 3 s.