STUPIDSID
1 (a)
State and prove Baye's theorem.
5 M
1 (b)
State the Axiomatic definition of probability
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1 (c)
If A and B are two events such that :P(A)=0.3,P(B)=0.4,P(A∩B)=0.2.
Find
(i)P(A∪B)
(ii)P((A/B)
(iii)P(A/B)
(iv)P(A∪B)
Find
(i)P(A∪B)
(ii)P((A/B)
(iii)P(A/B)
(iv)P(A∪B)
5 M
1 (d)
Explain the properties of distribution function
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2 (a)
The joint probability distribution of a two dimensional random variable (X,Y) is given by f(x,y)
=k x y e-(x2+y2); x ≥0, y ≥0. Find
(i) The value of k
(ii) Marginal density function of X and Y
iii) Conditional density function of Y given that X=x and Conditional density function of X given that Y=y
Check for independence of X and Y.
(i) The value of k
(ii) Marginal density function of X and Y
iii) Conditional density function of Y given that X=x and Conditional density function of X given that Y=y
Check for independence of X and Y.
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2 (b)
Explain moment generating function of discrete random variable and continuous random variable in
detail.
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3 (a)
If X,Y are two independent random variables with identical uniform distribution in(0,1), find the
probability density function of (U,V) where U=X+Y and V=X-Y. Are U and V independent.
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3 (b)
Find the characteristic function of Binomial distribution and Poisson distribution.
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4 (a)
Define Central limit theorem
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4 (b)
Describe the sequence of random variables.
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4 (c)
Explain and prove Chebychev's inequality
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5 (a)
A random process is given by x(t)=sin(Wt+Y) where Y is uniformly distributed over (0,2π) ,verify
whether {x(t)} is a wide sense stationary process
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5 (b)
State the properties of auto-correlation function and cross-correlation function.
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6 (a)
If a random process is given by x(t)=10cos(100t+θ) where θ is uniformly distributed over (-π,π) ,
prove that {x(t)} is correlation ergodic.
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6 (b)
A WSS random process {X(t)} is applied to the input of an LTI system whose impulse response is
te-at) u(t) where a(>0) is real constants. Find the mean of the output Y(t).
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7 (a)
State and prove Chapman-Kolmogorov equation.
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7 (b)
The transition matrix of Markov chain with three states 0,1 and 2 is given by
\[ P=\begin{matrix} 0\\ 1\\ 2 \end{matrix}\begin{bmatrix} 3/4 &1/4 &0 \\ 1/4&1/2 &1/4 \\ 0&3/4 &1/4 \end{bmatrix} \] and the initial state distribution is \[ P(x_o=i)=1/3, \ i=0,1,2 \\ Find :- \\ (i)\ P[X_2=2]\\ (ii)\ P[X_3=1, \ X_2=2, \ X_1=1, \ X_0=2] \]
\[ P=\begin{matrix} 0\\ 1\\ 2 \end{matrix}\begin{bmatrix} 3/4 &1/4 &0 \\ 1/4&1/2 &1/4 \\ 0&3/4 &1/4 \end{bmatrix} \] and the initial state distribution is \[ P(x_o=i)=1/3, \ i=0,1,2 \\ Find :- \\ (i)\ P[X_2=2]\\ (ii)\ P[X_3=1, \ X_2=2, \ X_1=1, \ X_0=2] \]
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