B.TECH - Semester 4 probability random processes and numerical techniques Question Paper 2019 (jun)
Practice authentic previous year university questions for better exam preparation.
Sample Questions
- 401 : PROBABILITY, RANDOM PROCESSES AND NUMERICAL TECHNIQUES (FR) Time : 3 Hours Max. Marks : 100 PART - A Answer all questions. Each question carries 4 marks. :
- If $f(x)=\frac{k}{2^{x}}$ is a probability distribution of a random variable which can take values $x=0,1,2,3,4$. Find $K$ and Mean of the distribution.
- Find the mean and variance of the probability distribution with density function $f(x)=K e^{-\frac{1}{8}\left(x^{2}+8 x+16\right)}$.
- The customers arrive at a bank according to a Poisson Process with mean rate of 2 minutes. Find the probability that during an 1 minute interval no customers arrive.
- The autocorrelation function of a stationary process $\{(X(t))\}$ is given by $R(\tau)=2+4 e^{-2|\tau|}$. Find mean and variance of the process $\{(X(t))\}$. P.T.O.
- Using Lagrange's interpolation formula find the value of $y$ when $x=9$ for the following data $$ \begin{array}{lccl} X & 5 & 6 & 11 \\ Y & 12 & 13 & 16 \end{array} $$ PART - B Answer one full question from each Module. Each question carries $\mat...
- (a) If $f(x)=\left\{\begin{array}{cc}0 & x<2 \\ \frac{1}{18}(2 x+3) & 2 \leq x<4 \\ 0 & x>4\end{array}\right.$ is the probability density function of a random variable. Find Mean and distribution function.
- (b) Human errors is given as the reason for $75 \%$ of all accidents in a plant. Use Binomial distribution to find the probability that human error will be given as the reason for 2 of the next 4 accidents.
- (c) The mean weight of 500 students at a certain school is 50 kg and the standard deviation is 6 kg . Assuming that the weights are normally distributed, find the expected number of students weighing
- (i) between 40 and 50 kg (ii) more than 60 kg .
- (a) A Random variable has the following probability distribution | $X$ | -2 | -1 | 0 | 1 | 2 | 3 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $f(x)$ | $1 / 10$ | $k$ | $1 / 5$ | $2 k$ | $3 / 10$ | $3 k$. | Find :
- (i) k (ii) Mean (iii) Variance (iv) $P(-2<X<2)$.
- (b) The number of cell phones sold daily in a shop is uniformly distributed with a minimum of 50 phones and a maximum of 100 phones. Find the probability that :
- (i) the daily sales will fall between 70 and 80 phones (ii) at least 75 phones are sold on a given day (iii) at most 70 phones are sold on a given day.
- (c) The monthly breakdown of a computer follows Poisson distribution with mean 1.2 . Find the probability that this computer will function for a month
- (i) without a break down (ii) with only one break down (iii) with at most two break down.
- (a) If $f(x)=\left\{\begin{array}{cc}e^{-(x+y)} & x \geq 0, y \geq 0 \\ 0 & \text { otherwise }\end{array}\right.$ is a joint probability density function of two dimensional random variable. Find $P\left(\frac{1}{2}<X<2,0<Y<4\right)$.
- (b) If $f(x, y)=2$ for $0<x<1,0<y<x$ is the joint probability density function of random variables $X$ and $Y$, find the marginal and conditional density functions. Are $X$ and $Y$ are independent?
- (c) Calculate the coefficient of correlation for the following data | $x$ | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $y$ | 15 | 16 | 14 | 13 | 11 | 12 | 10 | 8 | 9 |
- (a) Show that $X(t)=A \cos \left(\omega_{0} t+\theta\right)$ is WSS if $A$ and $\omega_{0}$ are constants and $\theta$ is Uniformly distributed in $(0,2 \pi)$ 10
- (b) Show that the Random Process $X(t)=A \cos \lambda t+B \sin \lambda t$ ( $A, B$ are Random variables) is WSS if $E(A)=E(B)=0, E\left(A^{2}\right)=E\left(B^{2}\right), E(A B)=0$.
- (a) If the auto correlation function of a random process is $R(\tau)=\rho e^{-\rho|\tau|}, \rho>0$, show that the spectral density is given by $S(w)=\frac{2}{1+\left(\frac{w}{\rho}\right)^{2}}$.
- (b) If the auto covariance function of a stationary process $X(t)$ is given by $C(\tau)=q e^{-\alpha|\tau|} \quad(\alpha>0$ and $q$ are constants). Show that $X(t)$ is mean ergodic. OR
- (a) If the auto correlation function of a WSS process is $R(\tau)=\rho e^{-\rho|\tau|}, \rho>0$, show that $X(t)$ is mean ergodic.
- (b) Suppose that customers arrive at a shop in accordance with a Poisson process with mean arrival of 5 minutes. Find the probability that during a time interval of 3 minutes
- (i) exactly 10 customers arrive (ii) more that 10 customers arrive.
- (a) Find the root between $(2,3)$ of $x^{3}-2 x-5=0$ by regulaFalsi method.
- (b) Solve by Gauss Seidal Iteration method $3 x+2 y=4.5,2 x+3 y-z=5,-y+2 z=-0.5$. Use Initial approximation $x_{0}=0.4, y_{0}=1.6, z_{0}=0.4$.
- (c) Evaluate $\int_{0}^{\frac{\pi}{2}} \sin x d x$ using:
- (i) Trapezoidal rule (ii) Simpson's rule with 10 equal intervals. 6 OR
- (a) Solve by Gauss Elimination method : $$ x+2 y+z=3,2 x+3 y+3 z=10,3 x-y+2 z=13 . $$
- (b) Using Newton's forward interpolation formula estimate $\sin 47^{\circ}$ given | $\theta$ | 45 | 50 | 55 | 60 | 65 | | :---: | :---: | :---: | :---: | :---: | :---: | | $\sin \theta$ | 0.7071 | 0.7660 | 0.8192 | 0.8660 | 0.9036 |
- (c) Using Newton-Raphson's method solve the equation $\cos x+1=3 x$ correct to 4 decimal places.