BE Computer Engineering (Pokhara University) Probability and Statistics (PU, MTH 216) Question Paper 2079

(a) State and prove Bayes' theorem for a sample space partitioned into mutually exclusive and exhaustive events $E_1, E_2, \dots, E_n$. (6)

(b) In an electronics assembly plant, three machines $M_1$, $M_2$ and $M_3$ produce 30%, 45% and 25% of the total output of microcontroller boards respectively. The proportion of defective boards produced by these machines is 2%, 3% and 4% respectively. A board is selected at random from the total output and is found to be defective.

(i) What is the probability that the selected board is defective?

(ii) Given that the board is defective, find the probability that it was produced by machine $M_2$.

(iii) Which machine is most likely to have produced the defective board? Justify your answer using the posterior probabilities. (10)

(a) Define a continuous random variable and state the properties that a probability density function (pdf) $f(x)$ must satisfy. (4)

(b) The lifetime (in thousands of hours) of an electronic component is a continuous random variable $X$ with pdf

(i) Determine the value of the constant $k$.

(ii) Find the mean and variance of the lifetime $X$.

(iii) Compute $P(1 \le X \le 3)$. (8)

(c) Distinguish between a binomial distribution and a Poisson distribution, stating one engineering situation where each is appropriate. (4)

(a) Explain the general procedure of testing a statistical hypothesis. Clearly define the terms null hypothesis, alternative hypothesis, Type I error, Type II error, level of significance, and critical region. (6)

(b) A manufacturer claims that the mean tensile strength of a certain type of wire is at least 250 MPa. A random sample of 36 wires gives a sample mean of 244 MPa with a sample standard deviation of 18 MPa.

(i) Formulate the appropriate null and alternative hypotheses.

(ii) Test the manufacturer's claim at the 5% level of significance.

(iii) State your conclusion and explain whether the manufacturer's claim is supported by the data. (10)

The marks obtained by 50 students in a programming examination are summarized below:

(a) Compute the mean and the median of the distribution.

(b) Calculate the standard deviation and the coefficient of variation, and comment on the consistency of the data.

The following data show the number of hours ($x$) eight students spent practising coding problems per week and their corresponding scores ($y$) in a competitive test:

(a) Fit the least-squares regression line of $y$ on $x$.

(b) Estimate the test score of a student who practises for 7 hours per week.

(c) Compute the Karl Pearson coefficient of correlation and interpret it.

The number of requests arriving at a web server follows a Poisson distribution with an average of 5 requests per minute.

(a) Find the probability that exactly 3 requests arrive in a given minute.

(b) Find the probability that at most 2 requests arrive in a given minute.

(c) Find the probability that more than 1 request arrives in a 30-second interval.

The diameters of ball bearings produced by a machine are normally distributed with a mean of 12.0 mm and a standard deviation of 0.04 mm. A bearing is acceptable if its diameter lies between 11.92 mm and 12.08 mm.

(a) Find the probability that a randomly selected bearing is acceptable.

(b) If 5000 bearings are produced, estimate how many will be rejected.

(c) Find the diameter exceeded by only 2.5% of the bearings.

(a) State the addition and multiplication theorems of probability for two events $A$ and $B$. (2)

(b) Two independent components $A$ and $B$ in a system have reliabilities (probabilities of functioning) 0.9 and 0.8 respectively. Find the probability that the system functions if the components are connected (i) in series, and (ii) in parallel. (4)

A random sample of 100 resistors drawn from a large production lot has a mean resistance of 102 ohms with a sample standard deviation of 8 ohms.

(a) Construct a 95% confidence interval for the true mean resistance of the lot.

(b) Interpret the meaning of this confidence interval.

(c) What sample size would be required to estimate the mean resistance within $\pm 1$ ohm at the same confidence level?

In a survey of software developers, 240 out of 400 respondents stated that they prefer working with statically typed programming languages. Test, at the 1% level of significance, whether the proportion of developers who prefer statically typed languages differs significantly from 0.5. State the hypotheses, compute the test statistic, and give your conclusion.

(a) Define skewness and kurtosis. Explain how they describe the shape of a frequency distribution. (3)

(b) For a moderately skewed distribution, the mean is 36, the median is 34, and the standard deviation is 6. Compute the Karl Pearson coefficient of skewness and comment on the nature of the distribution. (3)