Probability and Statistics for Data Science (AD-302) - Important Questions

Calculate the mean and standard deviation for a given grouped frequency distribution, including application to mean plus one/two standard deviations.

Define Data Science. Explain the Data Science life cycle.

Calculate arithmetic mean, median and mode for ungrouped raw data.

Define sampling distribution and explain the uses of sampling distributions.

Calculate descriptive statistics for the given PCO/STD customer count data.

Prove that for any discrete distribution the standard deviation is not less than the mean deviation from the mean.

The first of the two samples has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt{13.44}$, find the standard deviation of the second group.

Find the mean deviation from the mean and standard deviation of A.P. $a, a + d, a + 2d, \dots, a + 2nd$ and verify that the latter is greater than the former.

Explain the different components of statistics and its uses in everyday life with examples.

Explain that every average has its own peculiar characteristics and no single average is best, with examples.

Calculate mean life from cumulative failure data over time.

Given Bowley's coefficient of skewness, $Q_1$ and Median, find the coefficient of quartile deviation.

Given the first four moments about an arbitrary origin, compute the first four central moments about the mean and hence find $\beta_1$ and $\beta_2$ (and comment on skewness and kurtosis).

Calculate the first four central moments of a given frequency distribution and hence find $\beta_1$, $\beta_2$ / measures of skewness and kurtosis.

Using Bayes' theorem, given machine-wise output shares and defective rates, find the posterior probability that a randomly drawn defective bolt was manufactured by machine A, B, or C.

Given a frequency distribution, calculate the first four central moments about the mean and hence find $\beta_1$ and $\beta_2$.

If two dice are thrown, what is the probability that the sum is (i) greater than 8 (ii) neither 7 nor 11?

Find the moment generating function of the discrete binomial distribution given by

$P(x) = {^nC_x} p^x q^{n-x}$ where $q = 1-p$.

Also find the first and second moments about the mean.

If two dice are thrown, what is the probability that the sum is i) greater than 8 ii) neither 7 Nor 11?

Using Bayes' theorem, given three urns with specified compositions, find the posterior probability that white and red balls drawn after random choice of urn came from urn I, II, or III.

Find the probability that a random draw of 4 balls from a box with red, white and black balls contains at least one ball of each colour.

What is probability and how is it calculated under the classical approach?

Fit a second-degree parabola $y = a + bx + cx^2$ by the method of least squares to given $(x,y)$ data.

Using the Poisson distribution with mean $m$, compute Poisson probabilities such as $P(X=r)$, $P(X \ge k)$, or $P(X=0)$ over multiple units.

Fit a straight line / least-squares linear trend $y = a + bx$ to given $(x,y)$ data.

Ten coins are thrown simultaneously, find the probability of getting at least seven heads.

Given $X \sim N(\mu,\sigma^2)$, find $P(X \ge a)$ and $P(a < X < b)$ / percentage between limits.

Derive mean and variance of the Binomial distribution.

For continuous pdf $f(x)=cx^2$ on $0<x<1$, find normalizing constant $c$ and $P(1/3 < X < 1/2)$.

Define Binomial, Poisson and Normal distributions.

Prove that the Poisson distribution is the limiting case of the binomial distribution.

For normal income distribution, verify tail percentages and find cutoff for top $1\%$ / richest group.

Verify $f(x)=6x(1-x), 0\le x\le 1$ is a valid p.d.f. and find the median $b$ such that $P(X<b)=P(X>b)$.

Prove for normal distribution quartile deviation : mean deviation : standard deviation is approximately $10:12:15$.

Calculate Karl Pearson's coefficient of correlation between two variables from given bivariate data.

Calculate Spearman's rank correlation coefficient between two variables $X$ and $Y$ from given data.

Determine the two lines of regression from given discrete bivariate data.

Calculate pairwise rank correlation coefficients among three judges' rankings and determine which pair has the closest agreement.

For a given joint p.d.f. of X and Y, find the correlation coefficient, the two lines of regression, and the two regression curves for the means.

Estimate the most likely value of one variable corresponding to a given value of the other using means, standard deviations and correlation coefficient.

Examine whether given linear equations can be the estimated regression equations of Y on X and X on Y with theoretical justification.

Explain the meaning of regression and its role in business and economic analysis.

"Correlation and Regression are two sides of the same coin". Explain.

Single-sample t-test for population mean (test sample mean against hypothesized mean / compute t-statistic)

Test hypothesis about a single normal population variance using chi-square test on sample variance.

F-test for equality of two population variances from two independent samples

95% confidence interval for population mean from large sample

Two-sample large-sample test for significant difference between two population means

Test whether observed frequencies of digits follow a uniform distribution using chi-square goodness-of-fit test.

A continuous random variable $X$ has a p.d.f. $f(x) = 3x^2, 0 \le x \le 1$. Find $a$ and $b$ such that i) $P(X < a) = P(X > a)$, ii) $P(X > b) = 0.05$.

What is a sampling distribution, and what are the uses of sampling distributions?

The average monthly electricity consumption for a sample of $100$ families is $1250\text{ units}$. Assuming the standard deviation of electric consumption of all families is $150\text{ units}$, construct a $95\text{ percent}$ confidence interval estimate of the actual mean electric consumption.