EC-703-C Probability Theory and Stochastic Processor - Nov 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aState and prove Bays theorem of probability.
   How to answer

   • State Bayes theorem for partition B_i of sample space
   • Write conditional probability P(B_k|A) = P(A|B_k)P(B_k)/P(A)
   • Expand total probability P(A)=sum_i P(A|B_i)P(B_i)
   • Complete proof and mention applications
2. 7 Marks
   bIf A and B are independent events, prove that the events A' and B, A and B, and A and B' are also independent.
   How to answer

   • Assume P(A cap B)=P(A)P(B) for independent A,B
   • Use complement relations A' cap B = B - A cap B
   • Show P(A' cap B)=P(A')P(B) and similarly for others
   • Conclude all complement combinations remain independent
3. 7 Marks
   aA missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and B failing are known to be 0.01 and 0.03 respectively. It is also known that B is more likely to fail (probability 0.06) if A failed.

   1. i)What is the probability of an accidental missile launch?
   2. ii)What is the probability that A will fail if B has failed?
   3. iii)Are events "A fails" and "B fails" statistically independent.
   How to answer

   • Compute joint failure P(A cap B)=P(B|A)P(A)
   • Compute posterior P(A|B) using Bayes rule
   • Check independence condition P(A cap B) vs P(A)P(B)
   • Conclude on accidental launch probability
4. 7 Marks
   bA Gaussian random variable has mean value 1 and variance of 4. Find the probability that random variable has value between 1 and 2.
   How to answer

   • Standardize to Z=(X-mean)/sigma with mean 1 variance 4
   • Express P(1<X<2)=P(0<Z<0.5)
   • Use Q-function / Gaussian CDF table values
   • Compute numerical probability ~0.1915
5. 7 Marks
   aVerify that Rayleigh Density is a valid density Function.
   How to answer

   • Write Rayleigh pdf form with parameter sigma
   • Show f(x)>=0 for x>=0 and zero otherwise
   • Evaluate integral from 0 to infinity equals 1
   • Conclude it satisfies valid density conditions
6. 7 Marks
   bFind the moment generating function about origin of the Poisson distribution.
   How to answer

   • Write Poisson pmf P(X=k)=e^{-lambda}lambda^k/k!
   • Use definition M_X(t)=E[e^{tX}]=sum e^{tk}P(X=k)
   • Sum series to get exp[lambda(e^t-1)]
   • Mention mean=lambda from derivative at zero
7. 7 Marks
   aFind the density of the random variable $Z = X + Y$, where X and Y are two independent uniform random variables over $(-1, 1)$.
   How to answer

   • Write uniform pdfs f_X=1/2, f_Y=1/2 over (-1,1)
   • Use convolution f_Z(z)=int f_X(x)f_Y(z-x)dx
   • Evaluate triangular shape over (-2,2)
   • Give piecewise density: (2-|z|)/4
8. 7 Marks
   bDefine moment generating function. State and prove the properties of moment generating function.
   How to answer

   • Define M_X(t)=E[e^{tX}]
   • State properties: M(0)=1, derivatives give moments
   • Prove uniqueness and scaling/shift properties
   • Give example for common distributions
9. 7 Marks
   aState and prove the properties of Joint distribution function.
   How to answer

   • Define joint CDF F(x,y)=P(X<=x,Y<=y)
   • List non-negativity, limits 0 and 1, monotonicity
   • Prove rectangle probability >=0 and marginal relations
   • Conclude properties with proof steps
10. 7 Marks
    bExplain briefly about time average and ergodicity.
    How to answer

    • Define time average vs ensemble average
    • Define ergodicity: time average equals ensemble average
    • Give condition for mean and autocorrelation ergodic
    • Give example of ergodic process
11. 7 Marks
    aX(t) = ACoswt is a random process, where 'A' is uniform random variable over (0, $\pi$). Check X(t) for stationarity.
    How to answer

    • Write mean E[X(t)] using uniform pdf of A over (0,pi)
    • Compute autocorrelation E[X(t1)X(t2)]
    • Check if mean constant and R depends only on tau
    • Decide stationarity (SSS/WSS) conclusion
12. 7 Marks
    bConsider a random process $X(t) = \cos (\omega t + \theta)$ where $\omega$ is a real constant and $\theta$ is a uniform random variable in $(0, \pi/2)$. Show that X(t) is not a WSS process. Also find the average power in the process.
    How to answer

    • Compute mean with theta uniform over (0,pi/2)
    • Show mean is time-dependent hence not WSS
    • Compute autocorrelation and show depends on t1+t2
    • Find average power E[X^2(t)]
13. 7 Marks
    aProve that Power Spectral Density (PSD) and Auto correlation function of Random process form a Fourier transform pair.
    How to answer

    • Define PSD as Fourier transform of autocorrelation
    • Define autocorrelation R_XX(tau)=E[X(t)X(t+tau)]
    • Prove Wiener-Khinchin relations both directions
    • State conditions for existence
14. 7 Marks
    bProve that $S_{yy}(\omega)=|H(\omega)|^2 S_{xx}(\omega)$
    How to answer

    • Model output Y as convolution of X with h(t)
    • Take Fourier transform to get Y(omega)=H(omega)X(omega)
    • Compute output autocorrelation and its spectrum
    • Obtain S_yy=|H|^2 S_xx
15. 7 Marks
    aState and prove any three properties of cross power density spectrum.
    How to answer

    • Define cross PSD as Fourier transform of cross-correlation
    • Prove conjugate symmetry S_xy=S_yx*
    • Prove real part even, magnitude bound
    • Prove cross-correlation Fourier relation
16. 7 Marks
    bClassify random processes and explain.
    How to answer

    • List classification by time/state: continuous/discrete
    • Classify by stationarity: SSS, WSS, non-stationary
    • Classify by ergodicity and Markov/Poisson etc.
    • Explain each with example