STUPIDSID
1 (a)
State the Chebychev?s inequality and explain.
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1 (b)
What do you mean by steady state distribution of Markov chain.
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1 (c)
Suppose X and Y are two random variables when do you say that X and Y are
a) Orthogonal
b) Uncorrelated
a) Orthogonal
b) Uncorrelated
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1 (d)
What is the difference between a Random variable and a Random process?
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1 (e)
State and explain Baye's theorem.
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2 (a)
A certain test for a particular cancer is known to be 95% accurate. A person submits to the test
and the results are positive. Suppose that the person comes from a population of 1,00,000(one lakh) where 2000 people
suffer from that disease , what can we conclude about the probability that the person under test has that particular
cancer?
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2 (b)
Explain with suitable examples Continuous, Discrete and Mixed type random variable.
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3 (a)
Explain the concept of conditional probability and the properties of conditional
probability.
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3 (b)
Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white and 5 blue balls.
If we let X and Y denote respectively the number of red and white balls chosen.
Find :-
(i) The joint probability distribution of (X,Y)
ii) Probability mass function of X
(iii) Probability mass function of Y
Find :-
(i) The joint probability distribution of (X,Y)
ii) Probability mass function of X
(iii) Probability mass function of Y
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4 (a)
\[ Suppose \ f_X(X)=\dfrac {2X}{\pi^2}, 0<X< \pi \ and \ y=\sin x \ Determine \ f_y(y) \]
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4 (b)
Compare PDF of Binomials and Poison Random variable. A spacecraft has 1,00,000 components. The
probability of any one component being defective is 2×10-5. The mission will be in danger if five or more
components become defective. Find a probability of such an event.
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5 (a)
Define Central limit theorem and give its significance
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5 (b)
Describe the sequence of random variables.
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5 (c)
State and prove Chapman-Kolmogorov equation.
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6 (a)
Explain what do you mean by?
(i)Deterministic system
(ii) Stochastic system
(iii) Memory-less system.
Prove that if input to memory-less system is strict sense stationary(SSS) process x(t), the output y(t) is also SSS.
(i)Deterministic system
(ii) Stochastic system
(iii) Memory-less system.
Prove that if input to memory-less system is strict sense stationary(SSS) process x(t), the output y(t) is also SSS.
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6 (b)
If a random process is given by x(t)=100cos(100t+ θ) where θ is uniformly distributed over (-π,π)
, prove that {x(t)} is correlation ergodic.
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7 (a)
Explain power spectral density function. State its important properties and prove any one of the
property.
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7 (b)
Consider a random process x(t) that assumes the values=±1. Suppose that x(t)= ±1 with probability
1/2 and suppose that x(t) then changes polarity with each occurrence of an event in a poison process of rate α. Find the
mean, variance and Auto-Covariance of x(t).
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