BE Computer Engineering (IOE, TU) Probability and Statistics (IOE, SH 602 / ENSH 301) Question Paper 2079

The following data represent the breaking strength (in kg) of 60 samples of a cable produced in a factory:

(a) Compute the arithmetic mean, median and mode of the distribution. (b) Calculate the standard deviation and the coefficient of variation, and comment on the consistency of the production process. (c) Find the Karl Pearson's coefficient of skewness and interpret the shape of the distribution.

(a) State the axioms of probability. For any two events $A$ and $B$, prove the addition theorem $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. (b) Three machines $M_1$, $M_2$ and $M_3$ produce respectively 25%, 35% and 40% of the total output of a factory. The proportions of defective items produced by them are 5%, 4% and 2% respectively. An item is drawn at random from the total output and found to be defective. Using Bayes' theorem, find the probability that it was produced by machine $M_3$.

(a) Define the normal distribution and state its important properties. Show that for a normal distribution the mean, median and mode coincide. (b) The lifetime of a certain type of electronic component is normally distributed with mean 1200 hours and standard deviation 150 hours. (i) What proportion of components last more than 1450 hours? (ii) What proportion of components have a lifetime between 1000 and 1300 hours? (iii) If the manufacturer wishes to guarantee a minimum lifetime such that only 5% of components fail before the guarantee period, what should the guarantee period be? (Standard normal tables are attached.)

The following data show the number of hours studied ($X$) and the marks obtained ($Y$) by 8 students in an examination:

(a) Compute the Karl Pearson's coefficient of correlation between $X$ and $Y$ and interpret the result. (b) Obtain the two regression equations (the regression of $Y$ on $X$ and of $X$ on $Y$). (c) Estimate the marks obtained by a student who studies for 10 hours, and show that the geometric mean of the two regression coefficients equals the correlation coefficient.

A discrete random variable $X$ has the following probability distribution:

(a) Find the value of the constant $k$. (b) Compute the expectation $E(X)$ and the variance $\text{Var}(X)$. (c) Find $P(X \geq 2)$.

In a certain manufacturing process it is known that on average 4% of the items produced are defective. A random sample of 10 items is selected. Using the binomial distribution, find the probability that the sample contains (a) exactly 2 defective items, (b) at most 1 defective item, and (c) at least 1 defective item. Also state the mean and variance of the number of defectives in such a sample.

The number of telephone calls received at an exchange follows a Poisson distribution with an average of 3 calls per minute. (a) Find the probability that no call is received in a given minute. (b) Find the probability that more than 2 calls are received in a given minute. (c) Justify the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A random sample of 100 light bulbs taken from a large consignment gave a mean life of 1570 hours with a standard deviation of 120 hours. (a) Construct a 95% confidence interval for the true mean life of the bulbs in the consignment. (b) Explain the difference between a point estimate and an interval estimate, and state the properties of a good estimator.

A company claims that the mean tensile strength of its steel wire is 580 N/mm². A random sample of 64 wires gave a mean strength of 568 N/mm² with a standard deviation of 40 N/mm². (a) At the 5% level of significance, test whether the data provide sufficient evidence against the company's claim. (b) Clearly state the null and alternative hypotheses, and define Type I and Type II errors in this context.

A die is thrown 120 times and the following frequencies of the faces are observed:

Using the chi-square test of goodness of fit at the 5% level of significance, test whether the die is unbiased. State the degrees of freedom and your conclusion.

Write short notes on any THREE of the following: (a) Moments, skewness and kurtosis. (b) Mutually exclusive and independent events with examples. (c) Probability mass function versus probability density function. (d) Central Limit Theorem and its significance in sampling.