Introduction to Probability and Statistics (AL-302) - Important Questions

State Chebyshev's inequality and prove that the binomial distribution tends to Poisson distribution as $n \to \infty$, $p \to 0$ with $np = \lambda$ constant.

Define regression coefficient with properties and derive the moment generating function $M_X(t) = e^{\mu t + \frac{1}{2}\sigma^2 t^2}$ for $X \sim N(\mu,\sigma^2)$.

Given discrete probability mass function with parameter K over $x=0$ to $7$, determine K, evaluate $P(x<6)$ and $P(0\le x\le 4)$, find minimum K with $P(x\le K)>1/2$, determine distribution function, and find mean.

Define discrete random variable (and independence) and prove linearity of expectation $E(\sum X_i)=\sum E(X_i)$.

Prove that for a random variable $X$ and constants $a,b$, $\mathrm{Var}(aX+b) = a^2 \mathrm{Var}(X)$, in particular $\mathrm{Var}(X+k)=\mathrm{Var}(X)$ and $\mathrm{Var}(kX)=k^2\mathrm{Var}(X)$.

Determine (i) $P\left(\frac{B}{A}\right)$ (ii) $P\left(\frac{A}{B^C}\right)$ if A and B are events with $P(A)=\frac{1}{3}, P(B)=\frac{1}{4}, P(A \cup B)=\frac{1}{2}$

Given discrete probability mass function over $X=-3$ to $3$ with parameter K, find K, mean and variance.

Define moments and explain the types of moments.

Show that sum of two independent random variables follows normal distribution.

State and prove Chebyshev's inequality.

Find the mean/expected value of the sum of numbers when two dice are thrown.

Find the expected number of defective items when a sample is drawn without replacement.

For $f(x)=e^{-x},\ x\ge 0$, find $E(X)$, $E(X^2)$ and variance of $X$.

Define exponential distribution with parameter $\theta$ and obtain its mean, variance and moment generating function.

Given $f(x)=K(1-x^2)$ for $0<x<1$ (0 otherwise), find the normalizing constant K, mean and variance.

Define gamma distribution and obtain the mean and variance of an exponential distribution with parameter $\theta$.

Find the mean and standard deviation of a normal distribution given tail/percentile conditions.

Define gamma distribution and find its mean and variance.

For pdf $f(x)=\frac{\sin x}{2}$ on $[0,\pi]$, compute mean, mode, median and $P(0<X<\pi/2)$.

Given $f(x)=C e^{-|x|}$ for $-\infty<x<\infty$, show $C=1/2$ and find mean, variance, and $P(0<X<4)$.

If $X$ is a continuous random variable and $Y = aX+b$, prove $E(Y)=aE(X)+b$ and $V(Y)=a^2 V(X)$.

Explain exponential and gamma densities with properties.

Explain mean, median, mode, variance and mean deviation of a continuous random variable with pdf $f(x)$.

Derive the characteristic function $\varphi_X(t) = e^{i\mu t - \frac{1}{2}\sigma^2 t^2}$ for $X \sim N(\mu,\sigma^2)$.

Define bivariate distribution and explain its properties.

Using Bayes' theorem to find the posterior probability that an observed item (e.g., red ball, defective clock) came from a particular source (box/factory) given prior selection probabilities.

State and prove Bayes' theorem and apply it to find the probability that a randomly chosen color-blind person is male.

Describe bivariate distribution and give the probability density function of the bivariate normal distribution.

Define Bayes' theorem and apply it to find the probability of correct diagnosis given that the patient died.

Find / calculate Karl Pearson's coefficient of correlation from given bivariate data.

Poisson with mean 1.6 accidents per day, find probability of at least one and at most one accident.

Define correlation and explain its types and methods of studying correlation.

Write merits and demerits of measures of central tendency and find the median for grouped data.

Define Binomial distribution and obtain its mean and variance.

Calculate Spearman's rank correlation coefficient from given raw X and Y data (including tied values).

Find the regression line of $y$ on $x$ for given data.

Given the first four central moments $\mu_1, \mu_2, \mu_3, \mu_4$, calculate $\beta_1, \beta_2$ / skewness and kurtosis.

Given mean 16 and variance 8 of binomial $X$, find $P(X \ge 1)$ and $P(X > 2)$.

Poisson variate with variance 3, find $P(X=0)$, $P(0 < X \le 3)$ and $P(1 \le X < 4)$.

Binomial with 20% defectives and sample of 5, find probability of none defective, one defective, and $P(1 < X < 4)$.

Explain partial and multiple correlation coefficients.

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

Fit a straight line of the form $y = a + bx$ to the given data.

Test a hypothesis about a single population proportion using a large-sample Z-test.

Test whether one group mean is significantly greater than another using large-sample two-mean test with given sample sizes, means and standard deviations at 5% significance.

Test a large-sample mean against a hypothesized population mean and construct a 95% confidence interval for the population mean.

Define the normal distribution, state its properties, and describe the large-sample test procedure for difference of two means.

Fit the curve $y = ax + bx^2$ (parabola through origin with no constant term) to the given data by the method of least squares.

Test the significance of a large-sample mean against a hypothesized population mean.

Explain Chi-square test of goodness of fit and conditions for applying Chi-square test.

Define chi-square distribution and explain its properties.

Use $\chi^2$ test to test whether aircraft accidents are uniformly distributed over days of the week.

Use $\chi^2$ test to test independence of appointment/employment on basis of sex from given 2x2 data.

A sample of 26 bulbs gives a mean life of 900 hours with a standard deviation of 20 hours. The manufacturer claims that the mean life of bulbs is 1000 hours. Is the sample not up to the standard. (5% L.O.S.)

In one sample of 10 observations, the sum of the squares of the deviations of the sample values from mean was 120 and in the other sample of 12 observations, it was 314. Test whether the difference is significant at 5% level?