B.TECH - Semester 4 probability random processes and numerical techniques Question Paper 2020 (sep)
Practice authentic previous year university questions for better exam preparation.
Sample Questions
- Find the value of K if $f(x)=k(2-x), 0<x<2$ is a probability density function of a random variable.
- A random variable has uniform distribution over $(-3,3)$. Compute (a) $P[X<2]$
- A random variable has uniform distribution over $(-3,3)$. Compute (b) $P[|X|<2]$.
- If $X(t)$ is a WSS process with $E(X(t))=2$ and $R(\tau)=4+e^{-\frac{|\tau|}{10}}$ find the mean of $S=\int_{0}^{1} X(t) d x$.
- The autocorrelation function of a stationary process $\{(X(t))\}$ is given by $R(\tau)=\frac{25 \tau^{2}+36}{6.25 \tau^{2}+4}$ find mean and variance of the process $\{(X(t))\}$.
- Solve by Gauss elimination method $$ 2 x+3 y-z=5,4 x+4 y-3 z=3,2 x-3 y+2 z=2 . $$ Answer one full question from each module. Each question carries 20 marks :
- (a) If $f(x)=\left\{\begin{array}{cc}x & 0<x<1 \\ 2-x & 1 \leq x<2\end{array}\right.$ is the probability density function of a random variable. Find mean and distribution function.
- (b) If $5 \%$ of the electric bulbs manufactured by a company are defective. Use Poisson Distribution to find the probability that in a sample of 100 bulbs
- (i) None is defective (ii) 5 bulbs are defective.
- (c) In an examination 30\% of the candidates obtained marks below 40 and 10\% of the candidates got above 75 marks. Assuming that the marks are normally distributed . find the mean and standard deviation of the distribution.
- (a) A Random variable has the following probability distribution $$ \begin{array}{ccccc} X & 0 & 1 & 2 & 3 \\ f(x) & \mathrm{k} / 2 & \mathrm{k} / 3 & \frac{k+1}{3} & \frac{2 k+1}{3} \end{array} $$ Find (i) $k$ (ii) Mean (iii) variance (iv) Distrib...
- (b) The amount of time that a surveillance camera will run without having to be retest is a random variable having the exponential distribution with mean 120 days. Find the probability that such a camera will
- (i) Have to be retested in less than 24 days (ii) Not have to be retested at least 120 days.
- (c) A box contains 100 cell phones, 20 of which are defective. 10 cell phones are selected for inspection. Find the probability that
- (i) at least one is defective (ii) at most three are defective (iii) none of them are defective (iv) all of them are defective.
- (a) If $f(x, y)=K e^{-(2 x+y)} x \geq 0, y \geq 0$ is a joint probability density function of two dimensional random variable. Find
- (i) K (ii) Conditional Distributions of X and Y .
- (b) The joint probability density function of X and Y is given by $f(x)=(1-x)(1-y)$ for $0<x<1,0<y<1$. Find $P\left(0<x<\frac{1}{2}, \frac{1}{2}<y<1\right)$ prove that $X$ and $Y$ are independent.
- (c) Calculate the coefficient of correlation for the following data. $$ \begin{array}{lllllllll} \mathrm{X} & 22 & 26 & 29 & 30 & 31 & 31 & 34 & 35 \\ \mathrm{Y} & 20 & 20 & 21 & 29 & 27 & 24 & 27 & 31 \end{array} $$
- (a) If $X(t)=\sin (\omega t+Y)$ where $\omega$ is a constant and $Y$ is uniformly distributed on $(0,2 \pi)$. Prove that $X(t)$ is WSS.
- (b) If $X(t)$ and $Y(t)$ are independent zero mean WSS process and $Z(t)=X(t)+Y(t)$, Show that $Z(t)$ is WSS.
- (a) If the Power Spector density of a WSS process is a $S(\omega)=\left|\begin{array}{cc}\frac{b}{a}(a-|\omega|) & |\omega| \leq a \\ 0 & |\omega|>a\end{array}\right|$ where $a$ and $b$ are constants, find the auto correlation function of the process...
- (b) Find the average power of the random process $\{(X(t))\}$ if the Power Spectral Density is given by $S(w)=\frac{10 w^{2}+35}{\left(w^{2}+4\right)\left(w^{2}+9\right)}$. OR K-4249
- (a) If the auto correlation function of a WSS process is $R(\tau)=\rho e^{-\rho|\tau| e>0}$ Find the Powerspectral density.
- (b) If the customers arrive at a counter in accordance with Poisson Process with mean rate of 2 per minutes. Find the probability that the interval between 2 consecutive arrivals is
- (i) more than 1 minutes (ii) between 1 and 2 minutes (iii) 4 minutes or less. Module - IV
- (a) Using Newton Raphson method to solve the equation $x^{3}+x-1=0$ correct to 4 decimalplaces.
- (b) Using Lagrange's interpolation formula find the value of $y$ when $x=10$ for the following table | X | 5 | 6 | 9 | 11 | | :--- | :---: | :---: | :---: | :--- | | Y | 12 | 13 | 14 | 16 |
- (c) Evaluate $\int_{0}^{6} \frac{1}{1+x^{2}} d x$ using
- (i) Trapezoidal rule (ii) Simpson's rule with 6 equal intervals. OR
- (a) Solve by Gauss Siedel Iteration Method $$ 28 x+4 y-z=32,2 x+17 y+4 z=35, x+3 y+10 z=24 . $$
- (b) Construct the Newton's forward interpolation polynomial which takes the following values $$ \begin{array}{ccccc} \mathrm{X} & 0 & 1 & 2 & 3 \\ \mathrm{Y} & 1 & 2 & 1 & 10 \end{array} $$
- (c) Find the root of $x^{3}-x-11=0$ that lies between 2 and 3 correct to two decimal places by bisection method.