SIMG-713 SPR ING 2002REVIEW TOPICS F OR FINAL EXAMOutline of Topics1. Concept of an experiment that produces outcomes that can be different on each trial.2. Concept of events as sets of possible experiment outcomes.3. Probability as a number associated with events.(a) Properties of probability(b) Joint probabilities(c) Conditional probabilities(d) Statistical independence(e) Mutually exclusive events4. Random variables5. Av erages over random variables.(a) Computation of a verages and moments(b) Physical interpretation of mean and variance(c) Correlation and covariance6. Av erages over functions of a random variable7. Modeling of discrete events and random arrivals(a) Binomial distribution(b) Poisson distribution(c) Normal distribution.(d) Error function and z-score.(e) Normal approximation to binomial.8. Application to detector arrays9. Transfer function modeling(a) Mean value of output as a function of q.(b) Variance of output as a function of q.(c) Detective Quantum Efficiency(d) Detection of input level change - SNR10. Random Process(a) Ensemble of sample functions(b) Ensemble averages1. Mean value2. Autocorrelation and crosscorrelation3. Variance4. Function of random variable1(c) Stationary vs nonstationary(d) Wide-sense stationary.11. Parameter Estimates(a) Why is an estimate based on a sample function always a random variable?(b) What is an estimator (as opposed to an estimate)?(c) Whatisbiasasitrelatestoanestimator?(d) How does the mean and variance of an estimator of the mean change with the number of samplesused?12. Correlation, Covariance and Crosscorrelation(a) DeÞnition of covariance.(b) DeÞnition of correlation.(c) DeÞnition of crosscorrelation.(d) Correlation coefficient.(e) Bivariate normal distribution.13. Power spectrum(a) DeÞnition as transform of autocorrelation function - Wiener Khin tchine Theorem(b) Relationship to Fourier transform of a sample function(c) Describe the periodogram algorithm for spectral estimation.(d) Describe the correlogram algorithm for spectral estimation.14. Digital Filter Models(a) Difference equation(b) Block diagram(c) Impulse response(d) System function(e) Relationships between models15. Special Topics (Not for Þnal exam)(a) Z Transform1. DeÞnition of z-transform of a sequence2. Interpretation of z as a delay operator3. Relationship between input, output and system function in z-domain(b) What information does the substitution X(z) with z = eiωprovide?1. What range of values ofωis available?2. How do we relate to time or frequency for a time-sampled application?3. How does X(ω) for a function x(t) relate to X(eiω) for a discrete function ?(c) Generation of random sequences(d) How to simulate a bandlimited random process2Questions from old exam s1. Random variablesXandYhave a joint discrete probability functionPXY(X = m, Y = n)shown inthe table below.X\Y121 0.10.220.30.4(a) Find the mean value of X.(b) Find the variance of X.(c) Find the correlation coefficient ρXY.(d) Find the expected value of the function g (X, Y )=(X − Y )(X + Y )2. A random number generator can produce a sequence of samples Xn,n=1, 2,...,N.(a) How would you determine an approximation to the probability density function fX(x)? Describean algorithm that you would use to construct the approximation. List any parameters associatedwith the algorithm.(b) Provide an estimator for the mean value ofXand determine the variance of the estimate itpro vides under the assumption that the samples are statistically independent.3. A physial analysis of the efficiency of a photon detector has shown that 80% of the incoming photonscause an electron to be produced inside the detector. Construct a model for the detector output Yin terms of the detector input X and derive a formula for the DQE based on this model. List anyassumptions you use in constructing the model and deriving the DQE formula.4. Random variablesXandYtake on the values {x1,x2,x3,x4} and {y1,y2,y3} .The values forP [X = xi]andP [Y = yj|X = xi]are given in the table below. Note that some of the information, in the positionslabeled A, B, C, D, is missing.X x1x2x3x4P [X = xi] 0.1 0.2 A 0.3P [Y = y1|X = xi] B 0.25 0.25 0.25P [Y = y2|X = xi] 0.35 0.45 0.25 CP [Y = y3|X = xi] 0.4 0.3 D 0.25(a) Determine the values that should go into locationsA, B, C, Din the table.(b) Compute P [X = x2|Y = y3] .5. We have four boxes. Box 1 contains 2000 components of which 5% are defective. Box 2 contains 500components of which 40% are defective. Boxes 3 and 4 contain 1000 each with 10% defective. Weselectat randomone of the boxes and removeat randoma single component.(a) What is the probability that the selected component is defective?(b) We examine the defective component and Þnd it defective. On the basis of this evidence, what isthe probability it was drawn from Box 2?6. Two random processes x (n) and y (n) are related b yy (n)=nXk=−∞x (k) h (n − k)whereP∞k=0h2(k) is bounded.3(a) Find an expression for the cross-correlation between thexandysequences in terms of the impulseresponse function h (n) under the condition that x is a white-noise sequence with µx=0andvariance σ2x.(b) Find an expression for the variance of theysequence in terms of the impulse response functionh (n) under the condition that x is a white-noise sequence with µx=0and variance σ2x.7. Let X(n) be a discrete random process that can assume two values 0 and 1, with probabilities 0.2 and0.8, respectively. The process is ergodic. The valuesX(n)andX(m)are statistically independent ifm 6= n. This random process is the input to a digital Þlter with impulse responseh(n)=anstep(n)where a is a real number with |a| < 1. The output of the digital Þlter is a random process Y (n).(a) Determine whether Y (n) is a wide-sense stationary random process.(b) Find the expected v alue of Y (n) in terms of a.(c) Find the variance of Y (n) in terms of a8. A calibrated photon source is designed to produce photons at an average rate of λ0= 1000 photonsper second. The sources are tested after they have been manufactured to see if they are acceptable. Asource is acceptable if the actual rate λ satisÞes |λ − λ0| ≤ 5 photons per second. The acceptance testis to count the number N of photons in T =40seconds and accept the source if |N − 40, 000| < 200.(a) What is the probability that a source will be rejected by the test if it hasλ = λ0? You may wantto use the table of values of the normal distribution function that is attached.(b) What additional