Q: In the context of reliability theory, the reliability function $R(t)$ is defined as:

$R(t) = P(T > t)$, \,\text{where} \, T \, \text{is the lifetime of the system}$

Question 1

Which of the following is the probability density function (p.d.f.) of the Beta distribution of the first kind with parameters $m$ and $n$?

Accepted Answer

$f(x) = \frac{1}{B(m,n)} x^{m-1} (1-x)^{n-1}, \quad 0 \leq x \leq 1, \ m, n > 0$

Question 2

Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed random variables from a continuous distribution. What is the probability that the smallest observation is $X_1$ and the largest is $X_n$?

Accepted Answer

$\frac{2}{n!}$

Question 3

Which of the following statements about the method of maximum likelihood estimation (MLE) is correct?

Accepted Answer

MLEs maximize the likelihood function with respect to the parameter.

Question 4

A researcher is testing the mean of a normal population with known variance. Which test statistic should be used?

Accepted Answer

$Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}$

Question 5

In a one-way ANOVA, which assumption is NOT required?

Accepted Answer

The sample means are equal.

Question 6

Which of the following is a correct formula for the control limits of a $\bar{X}$-chart in Statistical Process Control, assuming the process standard deviation $\sigma$ is known?

Accepted Answer

$\mu \pm 3 \frac{\sigma}{\sqrt{n}}$

Question 7

In regression analysis, the coefficient of determination $R^2$ represents:

Accepted Answer

The proportion of variance in the dependent variable explained by the independent variable(s).

Question 8

Which of the following is a property of the exponential distribution?

Accepted Answer

It is memoryless: $P(X > s + t \mid X > s) = P(X > t)$

Question 9

A company wants to estimate the average lifetime of its product. Which sampling method is most appropriate if the population is naturally divided into non-overlapping groups and each group is internally heterogeneous?

Accepted Answer

Stratified random sampling

Question 10

In survival analysis, the hazard function $h(t)$ is best described as:

Accepted Answer

The instantaneous failure rate at time $t$, given survival up to time $t$

Question 11

Which of the following is a necessary condition for the unbiasedness of an estimator $\hat{	heta}$ of a parameter $	heta$?

Accepted Answer

$E[\hat{	heta}] = 	heta$

Question 12

A process is said to be in statistical control if:

Accepted Answer

All points on the control chart fall within the control limits and display no systematic pattern.

Question 13

In the context of order statistics, what is the expected value of the maximum of $n$ independent random variables uniformly distributed on $[0,1]$?

Accepted Answer

$\frac{n}{n+1}$

Question 14

Which of the following is NOT an assumption of the classical linear regression model?

Accepted Answer

The dependent variable is normally distributed.

Question 15

In a two-sample $t$-test for equality of means, which of the following is a correct null hypothesis?

Accepted Answer

$H_0: \mu_1 = \mu_2$

Question 16

Which of the following is the correct formula for the mean of a Beta distribution of the first kind with parameters $m$ and $n$?

Accepted Answer

$\frac{m}{m+n}$

Question 17

A manufacturer uses a $c$-chart to monitor the number of defects per item. Which distribution is most appropriate for modeling the number of defects?

Accepted Answer

Poisson distribution

Question 18

In the context of reliability theory, the reliability function $R(t)$ is defined as:

Accepted Answer

$R(t) = P(T > t)$, \,	ext{where} \, T \, 	ext{is the lifetime of the system}$

Question 19

Which of the following is a correct statement about stratified random sampling?

Accepted Answer

The population is divided into homogeneous subgroups and samples are drawn from each subgroup.

Question 20

In a regression model $Y = \beta_0 + \beta_1 X + \epsilon$, the least squares estimate of $\beta_1$ is given by:

Accepted Answer

$\hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$

Savitribai Phule Pune University – T.Y. B.Sc. Statistics Syllabus & Study Notes (Semester V & VI)

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