List of practice Questions

Let \( X_1, X_2, \dots, X_n \) be a random sample from a distribution with cumulative distribution function \( F(x) \). Let the empirical distribution function of the sample be \( F_n(x) \). The classical Kolmogorov-Smirnov goodness of fit test statistic is given by \[ T_n = \sqrt{n} D_n = \sqrt{n} \sup_{-\infty<x<\infty} | F_n(x) - F(x) |. \] Consider the following statements: \begin{enumerate} \item The distribution of \( T_n \) is the same for all continuous underlying distribution functions \( F(x) \). \item \( D_n \) converges to 0 almost surely, as \( n \to \infty \). \end{enumerate} Which one of the following statements is/are true?

Consider the following transition matrices \( P_1 \) and \( P_2 \) of two Markov chains: \[ P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 1/3 & 1/2 & 1/6 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad P_2 = \begin{bmatrix} 1/6 & 1/3 & 1/2 \\ 1/4 & 0 & 3/4 \\ 0 & 1 & 0 \end{bmatrix}. \] Which of the following statements is correct?

Let \( X_1, X_2, \dots, X_{20} \) be a random sample of size 20 from \( N_6(\mu, \Sigma) \), with det(\(\Sigma\)) \(\neq 0\), and suppose both \(\mu\) and \(\Sigma\) are unknown. Let \[ \bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad S = \frac{1}{19} \sum_{i=1}^{20} (X_i - \bar{X})(X_i - \bar{X})^T. \] Consider the following two statements: \begin{enumerate} \item The distribution of \( 19S \) is \( W_6(19, \Sigma) \) (Wishart distribution of order 6 with 19 degrees of freedom). \item The distribution of \( (X_3 - \mu)^T S^{-1} (X_3 - \mu) \) is \( \chi^2_6 \) (Chi-square distribution with 6 degrees of freedom). \end{enumerate} Then which of the above statements is/are true?

Let \( X_1, X_2, \dots, X_{18} \) be a random sample from the distribution \[ f(x; \theta) = \begin{cases} \frac{2x}{\theta} e^{-x^2/\theta}, & x>0,\\ 0, & x \leq 0. \end{cases} \] What is the maximum likelihood estimator (MLE) of \( \theta \)?

Let \( X_1, X_2, \dots, X_n \) be a random sample from a population \( f(x; \theta) \), where \( \theta \) is a parameter. Then which one of the following statements is NOT true?

A random sample \( X_1, X_2, \dots, X_6 \) is taken from a Bernoulli distribution with the parameter \( \theta \). The null hypothesis \( H_0: \theta = \frac{1}{2} \) is to be tested against the alternative hypothesis \( H_1: \theta>\frac{1}{2} \), based on the statistic \( Y = \sum_{i=1}^{6} X_i \). If the value of \( Y \) corresponding to the observed sample values is 4, then the p-value of the test is ________?

Let \(M\) be a 2 \(\times\) 2 real matrix such that \((I + M)^{-1} = I - \alpha M\), where \(\alpha\) is a non-zero real number and \(I\) is the 2 \(\times\) 2 identity matrix. If the trace of the matrix \(M\) is 3, then the value of \(\alpha\) is

Let \(\{X(t)\}_{t\geq 0}\) be a linear pure death process with death rate \(\mu_i = 5i\), \(i = 0, 1, \dots, N\), \(N \geq 1\). Suppose that \(p_i(t) = P(X(t) = i)\). Then the system of forward Kolmogorov’s equations is

Let \( S^2 \) be the variance of a random sample of size \( n>1 \) from a normal population with an unknown mean \( \mu \) and an unknown finite variance \( \sigma^2>0 \). Consider the following statements:
(I) \( S^2 \) is an unbiased estimator of \( \sigma^2 \), and \( S \) is an unbiased estimator of \( \sigma \).
(II) \( \frac{n-1}{n} S^2 \) is a maximum likelihood estimator of \( \sigma^2 \), and \( \sqrt{\frac{n-1}{n}} S \) is a maximum likelihood estimator of \( \sigma \).
Which of the above statements is/are true?

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a function defined by \[ f(x, y) = \begin{cases} \frac{x^2 y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Find the value of \( \frac{\partial f}{\partial x} \) at \( (0, 0) \).

In a steam power plant based on Rankine cycle, steam is initially expanded in a high-pressure turbine. The steam is then reheated in a reheater and finally expanded in a low-pressure turbine. The expansion work in the high-pressure turbine is 400 kJ/kg and in the low-pressure turbine is 850 kJ/kg, whereas the pump work is 15 kJ/kg. If the cycle efficiency is 32%, the heat rejected in the condenser is ________________ kJ/kg (round off to 2 decimal places).

An engine running on an air standard Otto cycle has a displacement volume of 250 cm\(^3\) and a clearance volume of 35.7 cm\(^3\). The pressure and temperature at the beginning of the compression process are 100 kPa and 300 K, respectively. Heat transfer during constant-volume heat addition process is 800 kJ/kg. The specific heat at constant volume is 0.718 kJ/kg.K and the ratio of specific heats at constant pressure and constant volume is 1.4. Assume the specific heats to remain constant during the cycle. The maximum pressure in the cycle is ________________ kPa (round off to the nearest integer).

A steady two-dimensional flow field is specified by the stream function \[ \psi = k x^3 y, \] where \( x \) and \( y \) are in meter and the constant \( k = 1 \, \text{m}^2 \text{s}^{-1} \). The magnitude of acceleration at a point \( (x, y) = (1 \, \text{m}, 1 \, \text{m}) \) is ________________ m/s² (round off to 2 decimal places).

Consider a solid slab (thermal conductivity, \( k = 10 \, \text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1} \)) with thickness 0.2 m and of infinite extent in the other two directions as shown in the figure. Surface 2, at 300 K, is exposed to a fluid flow at a free stream temperature (\( T_{\infty} \)) of 293 K, with a convective heat transfer coefficient (\( h \)) of 100 W\( \cdot\text{m}^{-2}\cdot\text{K}^{-1} \). Surface 2 is opaque, diffuse and gray with an emissivity (\( \varepsilon \)) of 0.5 and exchanges heat by radiation with very large surroundings at 0 K. Radiative heat transfer inside the solid slab is neglected. The Stefan-Boltzmann constant is \( 5.67 \times 10^{-8} \, \text{W}\cdot\text{m}^{-2}\cdot\text{K}^{-4} \). The temperature \( T_1 \) of Surface 1 of the slab, under steady-state conditions, is ________________ K (round off to the nearest integer).

During open-heart surgery, a patient’s blood is cooled down to 25°C from 37°C using a concentric tube counter-flow heat exchanger. Water enters the heat exchanger at 4°C and leaves at 18°C. Blood flow rate during the surgery is 5 L/minute.
Use the following fluid properties: \[ \begin{array}{|c|c|c|} \hline \text{Fluid} & \text{Density (kg/m}^3\text{)} & \text{Specific heat (J/kg-K)}
\hline \text{Blood} & 1050 & 3740
\text{Water} & 1000 & 4200
\hline \end{array} \] Effectiveness of the heat exchanger is ________________ (round off to 2 decimal places).

A thin-walled cylindrical pressure vessel has mean wall thickness of \( t \) and nominal radius of \( r \). The Poisson's ratio of the wall material is \( \frac{1}{3} \). When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1% and the corresponding wall thickness became \( \bar{t} \). The corresponding change in the wall thickness of the cylindrical portion, i.e. \( 100 \times \frac{\bar{t} - t}{t} \), is ________________% (round off to 3 decimal places).

A schematic of an epicyclic gear train is shown in the figure. The sun (gear 1) and planet (gear 2) are external, and the ring gear (gear 3) is internal. Gear 1, gear 3 and arm OP are pivoted to the ground at O. Gear 2 is carried on the arm OP via the pivot joint at P, and is in mesh with the other two gears. Gear 2 has 20 teeth and gear 3 has 80 teeth. If gear 1 is kept fixed at 0 rpm and gear 3 rotates at 900 rpm counter clockwise (ccw), the magnitude of angular velocity of arm OP is ________________ rpm (in integer).

Under orthogonal cutting condition, a turning operation is carried out on a metallic workpiece at a cutting speed of 4 m/s. The orthogonal rake angle of the cutting tool is \(5^\circ\). The uncut chip thickness and width of cut are 0.2 mm and 3 mm, respectively. In this turning operation, the resulting friction angle and shear angle are \(45^\circ\) and \(25^\circ\), respectively. If the dynamic yield shear strength of the workpiece material under this cutting condition is 1000 MPa, then the cutting force is ________________ N (round off to one decimal place).

A 1 mm thick cylindrical tube, 100 mm in diameter, is orthogonally turned such that the entire wall thickness of the tube is cut in a single pass. The axial feed of the tool is 1 m/minute and the specific cutting energy (\( u \)) of the tube material is 6 J/mm\(^3\). Neglect contribution of feed force towards power. The power required to carry out this operation is ________________ kW (round off to one decimal place).

A 4 mm thick aluminum sheet of width \( w = 100 \) mm is rolled in a two-roll mill of roll diameter 200 mm each. The workpiece is lubricated with a mineral oil, which gives a coefficient of friction, \( \mu = 0.1 \). The flow stress (\( \sigma \)) of the material in MPa is \( \sigma = 207 + 414 \varepsilon \), where \( \varepsilon \) is the true strain. Assuming rolling to be a plane strain deformation process, the roll separation force (\( F \)) for maximum permissible draft (thickness reduction) is ________________ kN (round off to the nearest integer).

Two mild steel plates of similar thickness, in butt-joint configuration, are welded by gas tungsten arc welding process using the following welding parameters.

A filler wire of the same mild steel material having 3 mm diameter is used in this welding process. The filler wire feed rate is selected such that the final weld bead is composed of 60% volume of filler and 40% volume of plate material. The heat required to melt the mild steel material is 10 J/mm³. The heat transfer factor is 0.7 and melting factor is 0.6. The feed rate of the filler wire is ________________ mm/s (round off to one decimal place).

An assignment problem is solved to minimize the total processing time of four jobs (1, 2, 3 and 4) on four different machines such that each job is processed exactly by one machine and each machine processes exactly one job. The minimum total processing time is found to be 500 minutes. Due to a change in design, the processing time of Job 4 on each machine has increased by 20 minutes. The revised minimum total processing time will be ________________ minutes (in integer).

The product structure diagram shows the number of different components required at each level to produce one unit of the final product P. If there are 50 units of on-hand inventory of component A, the number of additional units of component A needed to produce 10 units of product P is ________________ (in integer).

Consider a one-dimensional steady heat conduction process through a solid slab of thickness 0.1 m. The higher temperature side A has a surface temperature of 80°C, and the heat transfer rate per unit area to low temperature side B is 4.5 kW/m². The thermal conductivity of the slab is 15 W/m·K. The rate of entropy generation per unit area during the heat transfer process is ________________ W/m²·K (round off to 2 decimal places).

Consider steady, one-dimensional compressible flow of a gas in a pipe of diameter 1 m. At one location in the pipe, the density and velocity are 1 kg/m\(^3\) and 100 m/s, respectively. At a downstream location in the pipe, the velocity is 170 m/s. If the pressure drop between these two locations is 10 kPa, the force exerted by the gas on the pipe between these two locations is ________________ N.