JNTU
JNTU B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (August/September - 2022 - R18)
Answer any five questions. All questions carry equal marks.
Main
State and prove any four properties of joint distribution function.
Answer
The joint distribution function FXY(x, y) = P(X ≤ x, Y ≤ y) satisfies the following properties:
- FXY(∞, ∞) = 1. This follows because the probability of the entire sample space is 1.
- FXY(-∞, y) = 0 and FXY(x, -∞) = 0. Since the events {X ≤ -∞} or {Y ≤ -∞} are impossible, their joint probability is zero.
- FXY(x, y) is a non-decreasing function of both x and y. If x1 ≤ x2, then P(X ≤ x1, Y ≤ y) ≤ P(X ≤ x2, Y ≤ y).
- FXY(x, ∞) = FX(x) and FXY(∞, y) = FY(y). These represent the marginal distribution functions of X and Y respectively, obtained by letting the other variable approach infinity.
A Gaussian random variable has mean value 1 and variance of 4. Find the probability that random variable has value between 1 and 2.
Answer
Given X is a Gaussian random variable with mean μ = 1 and variance σ2 = 4, the standard deviation is σ = 2. We need to find P(1 < X < 2).
Standardizing the variable using Z = (X - μ) / σ:
Using the standard normal distribution table, Φ(0.5) ≈ 0.6915 and Φ(0) = 0.5. Therefore:
State and prove Bayes’s Theorem.
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Distinguish between mutually exclusive events and independent events.
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Consider the experiment of tossing four fair coins. The random variable X is associated with the number of tails showing. Compute and sketch the Cumulative Distribution Function (CDF) of X.
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Write the expressions for Rayleigh density and distribution function and list its characteristics.
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State Central limit theorem for the following two cases: i) Equal distributions ii) Unequal distributions.
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Let fXY(x, y) = 4x + 2y, 0 ≤ x ≤ 1/2, 0 ≤ y ≤ 1 and zero elsewhere find P (X≤ 1/4).
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State central limit theorem and explain its proof.
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The joint distribution of X, Y is given by fXY(X, Y) = 1/(4a2) * (1 + xy(x2 - y2)) for |X| ≤ a, |Y| ≤ a, a > 0, and 0 otherwise. Show that the Characteristic function of X+Y is equal to the product of the characteristic function of X and Y.
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Consider a random process X(t) = cos(ωt + θ) where ω is a real constant and θ is a uniform random variable in (0, π/2). Show that X(t) is not a WSS process. Also find the average power in the process.
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Determine if the constant process X(t) = A, where A is a random variable with mean A̅ and variance σA2 is mean ergodic.
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A WSS random process X(t) is applied to the input of an LTI system whose impulse response is 5t e-2t. The mean of X(t) is 3. Find the output of the system.
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State and prove properties of cross correlation function.
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Prove that Power Spectral Density (PSD) and Auto correlation function of Random process form a Fourier transform pair.
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Find the ACF of the following PSDs: i) Sxx(ω) = (157 + 12ω2) / ((16 + ω2)(9 + ω2)) ii) Sxx(ω) = 8 / (9 + ω2)2
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An amplifier has input and output impedances of 75 ohm, 60 dB power gain, and a noise equivalent bandwidth of 15 KHz. When a 75-ohm resistor at 290 K is connected to the input, the output rms noise voltage is 75 microvolt. Determine the effective noise temperature of the amplifier assuming that the meter is impedance matched to the amplifier.
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Discuss the necessity for “Source coding”.
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