B.TECH - Semester 4 engineering mathematics 3 probability and random processes Question Paper 2020 (sep)
Practice authentic previous year university questions for better exam preparation.
Sample Questions
- A continuous random variable $X$ has a pdf $f(x)=3 x^{2}, 0 \leq x \leq 1$, find $a$ and $b$ such that $P[X \leq a]=P[X>a]$ and $P[X>b]=0.05$.
- Find the mean and variance of the number of heads when 3 coins are tossed.
- If $X$ is uniformly distributed in $\left[-\frac{5}{4}, \frac{5}{4}\right]$, find $P\left[X<\frac{1}{2}\right]$.
- Find the mean and standard deviation of the normal distribution $f(x)=c e^{\frac{-\left(x^{2}-6 x+4\right)}{24}} ;-\infty<x<\infty$.
- The joint density of $X$ and $Y$ is given by $f(x, y)=\left\{\begin{array}{ll}k e^{-(2 x+3 y)} ; x>0, y>0 \\ 0, & ; \text { elsewhere }\end{array}\right.$. Find $k$. Are $X$ and $Y$ independent?
- Suppose that the customers are arriving at a ticket counter according to a Poisson process with mean rate of 2 per minute. Then in an interval of 5 minutes, find the probability that the number of customers arriving is (a) Exactly 4 (b) less than 3 .
- Find the variance of the stationary process $\{X(t)\}$, whose autocorrelation function is given by $R(\tau)=2+4 e^{-2|\tau|}$.
- If the transition probability matrix of a Markov chain is $\left[\begin{array}{cc}0 & 1 \\ 0.5 & 0.5\end{array}\right]$. Find the steady state distribution of the chain.
- Prove that the spectral density function of a real random process is an even function of $\omega$.
- Define (a) Time average of a Random process (b) Ergodicity. ( $\mathbf{1 0} \boldsymbol{\times} \mathbf{4} \boldsymbol{=} \mathbf{4 0}$ Marks) PART - B
- (a) If $2 \%$ of the electric bulbs manufactured by a company are defective, find the probability that in a sample of 100 bulbs (i) only 3 are defectives (ii) 5 or more defectives.
- (b) In a normal distribution 31\% of the items are under 45 and 8\% are above 64 . Find the mean and standard deviation.
- (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.
- (b) If $X(t)=r \cos (\omega t+\phi)$ where the random variables $r$ and $\phi$ are independent and $\phi$ is uniform is ( $-\pi, \pi$ ). Find $R\left(t_{1}, t_{2}\right)$.
- (a) If $X(t)=Y \cos t+Z \sin t$ where $Y$ and $Z$ are random variables, each of which assumes the values -1 and 2 with probabilities $2 / 3$ and $1 / 3$ respectively. Prove that $X(t)$ is WSS.
- (b) Derive the mean, variance and autocorrelation function of Poisson process. $10+10=20$
- (a) The transition probability matrix of a Markov process $\left\{X_{n}\right\}, n=1,2, \ldots$ having 3 states 0,1 and 2 is $P=\left[\begin{array}{ccc}3 / 4 & 1 / 4 & 0 \\ 1 / 4 & 1 / 2 & 1 / 4 \\ 0 & 3 / 4 & 1 / 4\end{array}\right]$. And the initia...
- (b) Find the power spectral density if $R(\tau)=e^{-\alpha|\tau|} \cos \beta \tau$. 10+10=20
- (a) If the power spectral density of a WSS process is given by $S(\omega)= \begin{cases}\frac{b}{a}(a-|\omega|) ; & |\omega| \leq a \\ 0 & ;|\omega|>a\end{cases}$ Find the autocorrelation function of the process.
- (b) Suppose that the probability of a dry day following a rainy day is $\frac{1}{3}$ and the probability of a rainy day following a dry day is $\frac{1}{2}$. Given that May 1 is a dry day.
- (i) What is the probability that May 3 is a rainy day? (ii) What is the probability that May 5 is a dry day? $\quad 10+10=20$