STUPIDSID
1 (a)
If A and B are any two events, then prove that P(A∪B)=P(A)+P(B)-P(A?B)
5 M
1 (b)
Explain the concept of conditional probability with an example
5 M
1 (c)
State and prove Baye's Theorem and Total Probability Theorem
10 M
2 (a)
Define discrete and continuous random variables, give one example of each type. Define exception of discrete random variable and continuous random variable
10 M
2 (b)
Suppose two million lottery tickets are issued with 100 winning tickets among them.
(i) If a person purchase 100 ticket, what is the probability of winning?
(ii) How many tickets should one buy to be 95% confident of having a winning ticket?
(i) If a person purchase 100 ticket, what is the probability of winning?
(ii) How many tickets should one buy to be 95% confident of having a winning ticket?
10 M
3 (a)
Find the characteristics function of Poisson distribution and find it's mean and variance
10 M
3 (b)
Let X be a random variable with CDF Fx(x) and PDF f(x)(x). Let y=aX+B where a and b are real constant and a ≠0. Fid PDF of y in terms of Fx(x)
10 M
4 (a)
Suppose that X and Y are continues random varibles with Joint Probability Density function
\[ f_{xy}(x,y)=\dfrac {xe^{-y}}{2}; \ \ 0<x<2, \ \ y>0 \] \[=0 \ \ elsewhere \]
\[ f_{xy}(x,y)=\dfrac {xe^{-y}}{2}; \ \ 0<x<2, \ \ y>0 \] \[=0 \ \ elsewhere \]
10 M
4 (b)
(ii) Find the join cumulative distribution function of X and Y
Find the marginal probability density functions of X and Y.
Find the marginal probability density functions of X and Y.
10 M
5 (a)
Define Central Limit Theorem and give its significance
5 M
5 (b)
Describe sequence of ramdom variables
5 M
5 (c)
IF two random variable are independent then prove that the density of their sum equals the convolution of their density functions.
10 M
6 (a)
Consider a random process X(t) defined by X(t)=A Cos(ωt+θ); -∞<t<∞ where A and ω are constant and θ is a uniform random variable over (-π, π). Show that X(t) is WSS
10 M
6 (b)
Prove that if the input to a linear time invaiant system is WSS then the output is also WSS
10 M
7 (a)
Explain power spectral density. State its important properties and prove any one property
10 M
7 (b)
State and prove the Chapman-Kolmogorov equation.
10 M
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