B.TECH - Semester 4 engineering mathematics probability and random process Question Paper 2020 (sep)
Practice authentic previous year university questions for better exam preparation.
Sample Questions
- A continuous random variable has a p.d.f $f(x)=k x^{2} e^{-x}, x \geq 0$. Find k and mean.
- The probability that a patient recovers from a disease is 0.4 . If 15 persons have such a disease, use suitable distribution to determine the probability that exactly 5 survive.
- An average scanned image occupies 0.6 megabytes of memory with a standard deviation of 0.4 megabytes. If you plan to install 80 images on your website, what is the probability that their total size is between 47 megabytes and 56 megabytes?
- Find the power spectral density of a WSS process with autocorrelation function $R(\tau)=e^{-\alpha \tau^{2}}$
- Prove that the Poisson process is a Markov process. P.T.O. PART - B Answer one full questions from each Module. Each question carries 20 marks.
- (a) Data was collected over a period of 10 years showing number of deaths from horse kicks in each of the 200 army corps. The distribution of deaths was as follows : No. of deaths : $\begin{array}{lllll}0 & 1 & 2 & 3 & 4 \\ \text { Total }\end{array...
- (b) The probability function of an infinite discrete distribution is given by $P(X=j)=\frac{1}{2^{j}},(j=1,2, \ldots, \infty)$. Verify that the total probability is 1 and find the mean and variance of the distribution. Also find $P(X \leq 5)$.
- (a) (i) A target is to be destroyed in a bombing exercise. There is $75 \%$ chance that any one bomb will strike the target. Assume that two direct hits are required to destroy the target completely. How many bombs must be dropped in order that the c...
- (b) If the density function of a continuous random variable X is given by $f(x)=\left\{\begin{array}{cc}a x, & 0 \leq x \leq 1, \\ a, & 1 \leq x \leq 2 \\ 3 a-a x, & 2 \leq x \leq 3 \\ 0, & \text { elsehwere }\end{array}\right.$
- (i) Find the value of a. (ii) find the cdf of x . (iii) if $x_{1}, x_{2}$ and $x_{3}$ are 3 independent observations of $x$, what is the probability that exactly one of these 3 is greater than 1.5 .
- (a) Let X and Y are two random variables having joint density function $f(x, y)=\left\{\begin{array}{cc}\frac{1}{8}(6-x-y), & \text { for } 0<x<2,2<y<4 \\ 0 & \text { otherwise } .\end{array}\right.$ Find
- (i) $P(X<1 \cap Y<3)$, (ii) $P(X+Y<3)$ and (iii) $P\left(X<\frac{1}{Y}<3\right)$.
- (b) Calculate the correlation coefficient for the following heights (in inches) of father X and their sons Y . $$ \begin{array}{lllllllll} \mathrm{X} & 65 & 66 & 67 & 67 & 68 & 69 & 70 & 72 \\ \mathrm{Y} & 67 & 68 & 65 & 68 & 72 & 72 & 69 & 71 \end{...
- (a) The process $\{X(t)\}$ has the distribution $P\{X(t)=n\}= \begin{cases}\frac{(a t)^{n-1}}{(1+a t)^{n+1}} & n=1,2 \\ \frac{a t}{(1+a t)}, & n=0\end{cases}$ Prove that the process is not stationary.
- (b) If $\{\mathrm{X}(\mathrm{t})\}$ is a WSS process with autocorrelation function $R_{x x}(\tau)$ and if $Y(t)=X(t+a)-X(t-a)$. Show that $R_{y y}(\tau)=2 R_{x x}(\tau)-R_{x x}(\tau+2 a)-R_{x x}(\tau-2 a)$.
- (a) Given that a process $\{X(t)\}$ has the autocorrelation function $R_{x x}(\tau)=A e^{-\alpha|\tau|} \cos \left(w_{0} \tau\right)$, where $A>0, \alpha>0$ and $\omega_{0}$ are real constants, find the power spectrum of $\{X(t)\}$.
- (b) Prove that the spectral density and auto correlation function of real WSS process form a Fourier cosine transform.
- (a) A man is at the integral part of the $x$-axis between the origin and the point $x=3$. He takes r unit step to the right with probability $1 / 3$ or to the left with probability $2 / 3$ unless he is at the origin, when he takes a step to the right...
- (i) he is at the point 1 after 3 walk and (ii) he is at 1 after a long run.
- (b) Prove that the random process $\{X(t)\}=A \cos (\omega t+\theta)$ where $A$ and $\omega$ are constants and $\theta$ is a uniformly distributed random variable in ( $0,2 \pi$ ) is correlation ergodic.
- (a) (i) Prove that the inter-arrival time of a Poisson process with parameter $\lambda$ has an exponential distribution with mean $\frac{1}{\lambda}$. (ii) If $\left\{\mathrm{N}_{1}(\mathrm{t})\right\}$ and $\left\{\mathrm{N}_{2}(\mathrm{t})\right\}$...
- (b) If $\{X(t)\}$ is a Gaussian process with $\mu(t)=10$ and $C\left(t_{1}, t_{2}\right)=16 e^{-\left|t_{1}-t_{2}\right|}$ find the probability that
- (i) $X(10) \leq 8$, and (ii) $|X(10)-X(6)| \leq 4$.
- (a) Obtain the steady state probabilities of the model ( $M / M / c$ ) with infinite capacity and also derive the average number of customers in the system and in the queue.
- (b) Arrivals at a telephone booth are considered to be Poisson with an average time of 12 minutes between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with mean 4 minutes.
- (i) Find the average number of persons waiting in the system. (ii) What is the probability that a person arriving at the booth will have to wait in the queue? (iii) What is the probability that it will take him more than 10 minutes altogether to wait...
- (v) The telephone department will install a second booth, when convinced that an arrival has to wait on the average for at least 3 minutes for phone. By how much the flow of arrivals should increase in order to justify a second booth? (vi) What is th...