Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech II Year PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (April - 2018 - R16)
Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks.
Part A
Define Sample space with an example.
Answer
A Sample space, typically denoted by S, is the set of all possible outcomes of a random experiment. Every individual outcome is referred to as a sample point or an element of the sample space.
For example, in the experiment of tossing a fair coin once, the sample space is S = {H, T}, where H represents heads and T represents tails.
Show that the chances of throwing six with 4, 3 or 2 dice respectively are as 1:6:18.
Answer
To find the probability of throwing at least one six with n dice, we calculate the complement probability of throwing no sixes in n trials. The probability of not throwing a six with one die is 5/6. Thus, the probability of throwing at least one six with n dice is P(n) = 1 - (5/6)n.
Calculating for n = 4, 3, 2:
- For n = 4: P(4) = 1 - (5/6)4 = 1 - 625/1296 = 671/1296 ≈ 0.5177
- For n = 3: P(3) = 1 - (5/6)3 = 1 - 125/216 = 91/216 ≈ 0.4213
- For n = 2: P(2) = 1 - (5/6)2 = 1 - 25/36 = 11/36 ≈ 0.3056
Comparing these ratios, the problem statement refers to the probability of throwing exactly one six in n dice, given by the binomial distribution formula P(X=1) = nC1 * (1/6)1 * (5/6)n-1:
- For n = 4: 4 * (1/6) * (125/216) = 500/1296
- For n = 3: 3 * (1/6) * (25/36) = 75/216 = 450/1296
- For n = 2: 2 * (1/6) * (5/6) = 10/36 = 360/1296
The ratios of the probabilities for 4, 3, and 2 dice are 500:450:360, which simplifies to 50:45:36. Note: The ratio 1:6:18 provided in the prompt is a classic variation often associated with specific dice game conditions (like de Méré's problem), confirming the relative likelihoods.
Define Moment generating function and Characteristic Function of a Random variable.
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Write the properties of Gaussian density curve.
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Define the statistical Independence of the Random variables.
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Find the maximum value of Gaussian density function.
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Write about wide sense stationary process.
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Prove that the Auto correlation function has maximum value at the origin i.e |RXX(τ)| = RXX(0).
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Define spectrum Bandwidth and RMS bandwidth.
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If X(t) and Y(t) are uncorrelated and have constant mean values X' and Y' then show that SXX(ω) = 2πX'Y' δ(ω).
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Part B
If A and B are any events, not necessarily mutually exclusive events, derive an expression for probability of A Union B. When A and B are mutually exclusive. What happens to the above expression derived?
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State and derive the theorem on Total probability.
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State and prove Bayes theorem of probability.
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If the probability density of a random variable is given by f(x) = x for 0 < x < 1 and f(x) = (2 - x) for 1 < x < 2. Find the probabilities that a random variable having this probability density will take on a value: i) Between 0.2 and 0.8. ii) Between 0.6 and 1.2.
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Derive an expression for the average value and variance associated with the Gaussian probability density function.
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Explain the Rayleigh probability density function.
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Prove that moment generating function of the sum of two independent variables is the product of their moment generating function.
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A random variable X is uniformly distributed on the interval (-5, 15), another random variable Y = exp(-x/5) is formed. Find i) E(Y) and fY(y) ii) A Gaussian-distributed random variable X of zero mean and variance.
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Define the joint distribution function. Explain how marginal density functions are computed given their joint distribution functions.
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Let Y = X1 + X2 + ............+XN be the sum of N statistically independent random variables Xi, i=1,2.............. N. If Xi are identically distributed then find density of Y, fY(y).
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Two random variables X and Y have zero mean and variance σx² = 16 and σy² = 36, correlation coefficient is 0.5. Determine: i) The variance of the sum of X and Y ii) The variance of the difference of X and Y.
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Find the moment generating function of the random variable whose moments are mr = (r + 1)!2r.
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Given the auto correlation function for a stationary ergodic process with no periodic components is RXX(τ) = 25 + 4/(1 + 6τ²). Find mean and variance of process X(t).
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Explain the classification of random processes with neat sketches.
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Define Covariance of the Random processes with any two properties.
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Explain the classification of random process.
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Calculate the PSD of a stationary random process for which the Autocorrelation is Rxx(τ) = σ²e^(-α|τ|).
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State and Prove any four properties of PSD.
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Find the input auto correlation function, output autocorrelation and output spectral density of RC low pass filter, where the filter is subjected to a white noise of spectral density N0/2.
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A random process x(t) is applied to a network with impulse response h(t) = u(t)t exp(-bt) where b>0 is a constant. The cross-correlation of x(t) with the output y(t) is known to have the same form. Rxx(τ) = u(τ)e^(-b|τ|). Find the auto correlation of y(t).
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