KERALA UNIVERSITY Class 4 engineering mathematics 3 probability and random processes Question Paper 2020

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Sample Questions

(c) Fit a binomial distribution to the following data : $7+7+6=20$ | $x:$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $f:$ | 2 | 11 | 19 | 30 | 14 | 8 | 5 | 1 |
(a) If $f(x, y)=\left\{\begin{array}{l}2 ; 0<x<1,0<y<x \\ 0 ; \text { otherwise }\end{array}\right.$ is a joint pdf, find the marginal and conditional density functions. Are $X$ and $Y$ statistically independent?
(b) If $X$ is uniformly distributed in $(-\sqrt{3}, \sqrt{3})$, then use Chebyshev's inequality to find an upper bound for $P(|x| \geq 3 / 2)$. Also find the exact probability $P(|X| \geq 3 / 2)$.
(c) Find $P(0.45<\bar{X}<0.55)$ using Central Limit Theorem, given that $\bar{X}$ denote the mean of a random sample of size 75 taken from the distribution with pdf $f(x)=\left\{\begin{array}{l}1 ; 0<x<1 \\ 0 ; \text { otherwise }\end{array}\right.$.
(a) Explain the classification of Random process with suitable examples.