AL-302 Introduction to Probability and Statistics - Dec 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aDefine discrete random variable and independence of random variable. Also, show that $$E(X_1 + X_2 + \dots + X_n) = E(X_1) + E(X_2) + \dots + E(X_n)$$
   How to answer

   • Define discrete random variable with countable support and PMF.
   • Define statistical independence of random variables: $P(X=x, Y=y) = P(X=x)P(Y=y)$.
   • State definition of expectation $E[X] = \sum x P(X=x)$.
   • Expand $E[X_1 + \dots + X_n]$ using joint probability distribution.
   • Rearrange summations over marginals to arrive at $\sum E[X_i]$.
2. 7 Marks
   bDefine expectation of random variables. Also, show that $$V(aX + b) = a^2 V(X)$$
   How to answer

   • Define mathematical expectation for discrete and continuous random variables.
   • State expression for $E[aX + b] = a E[X] + b$.
   • Apply definition of variance: $V(aX + b) = E[((aX + b) - E[aX + b])^2]$.
   • Simplify inner term: $(aX + b - a E[X] - b) = a(X - E[X])$.
   • Factor out $a^2$ using linearity of expectation to get $a^2 V(X)$.
3. 7 Marks
   aState Chebyshev's inequality. If $n \to \infty$, $p \to 0$ and $np = \lambda$, then show that binomial distribution reduces to Poisson distribution.
   How to answer

   • State Chebyshev's inequality: $P(|X - \mu| \ge k\sigma) \le 1/k^2$ or $P(|X - \mu| < k\sigma) \ge 1 - 1/k^2$.
   • Write binomial PMF: $P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$.
   • Substitute $p = \lambda/n$ and write $\binom{n}{x} = \frac{n(n-1)\dots(n-x+1)}{x!}$.
   • Take limit as $n \to \infty$ of individual factors: $\frac{n(n-1)\dots(n-x+1)}{n^x} \to 1$ and $(1 - \lambda/n)^n \to e^{-\lambda}$.
   • Conclude with Poisson PMF: $\frac{e^{-\lambda} \lambda^x}{x!}$.
4. 7 Marks
   bDefine gamma distribution. If $X$ follows exponential distribution with parameter $\theta$, then obtain its mean and variance.
   How to answer

   • Define gamma distribution with PDF $f(x) = \frac{\alpha^p}{\Gamma(p)} x^{p-1} e^{-\alpha x}$ for $x > 0$.
   • State PDF of exponential distribution: $f(x) = \theta e^{-\theta x}$ for $x \ge 0$.
   • Evaluate mean $E[X] = \int_0^\infty x \theta e^{-\theta x} dx = 1/\theta$.
   • Evaluate $E[X^2] = \int_0^\infty x^2 \theta e^{-\theta x} dx = 2/\theta^2$.
   • Compute variance $V(X) = E[X^2] - (E[X])^2 = 2/\theta^2 - 1/\theta^2 = 1/\theta^2$.
5. 7 Marks
   aDefine regression coefficient with its properties. If $X \sim N(\mu, \sigma^2)$, then show that $M_X(t) = e^{\mu t + \frac{1}{2}\sigma^2 t^2}$.
   How to answer

   • Define regression coefficients $b_{yx}$ and $b_{xy}$ as slopes of regression lines.
   • List key properties: $r = \pm \sqrt{b_{yx} b_{xy}}$, both have the same sign as $r$, arithmetic mean is $\ge r$, independent of change of origin but not scale.
   • Set up integral for MGF of normal distribution: $M_X(t) = \int_{-\infty}^\infty e^{tx} \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)} dx$.
   • Substitute $z = (x-\mu)/\sigma$ and complete the square in exponent: $t\mu + t\sigma z - z^2/2 = \mu t + \frac{1}{2}\sigma^2 t^2 - \frac{1}{2}(z - \sigma t)^2$.
   • Integrate Gaussian kernel to obtain $M_X(t) = e^{\mu t + \frac{1}{2}\sigma^2 t^2}$.
6. 7 Marks
   bDefine normal distribution with its properties. Describe the methodology for difference of means for large samples.
   How to answer

   • Define normal distribution with PDF $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$.
   • State key properties: bell-shaped, symmetrical about mean, unimodal, mean=median=mode, total area = 1, inflection points at $\mu \pm \sigma$.
   • State null hypothesis $H_0: \mu_1 = \mu_2$ and alternative hypothesis $H_1: \mu_1 \ne \mu_2$ (or one-tailed).
   • Give standard error of difference between sample means: $SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$.
   • State test statistic $Z = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\sigma_1^2/n_1 + \sigma_2^2/n_2}}$ and decision rule at significance level $\alpha$.
7. 7 Marks
   aWhat do you understand by measure of central tendency? Also, write down its merits and demerits. Calculate mean and standard deviation for the observations 5, 10, 20, 25, 40, 42, 45, 48, 70, 80.
   How to answer

   • Define measures of central tendency (mean, median, mode) as single representative values of data.
   • List merits (rigidly defined, based on all observations) and demerits (affected by extreme values).
   • Sum observations: $\sum x_i = 5 + 10 + 20 + 25 + 40 + 42 + 45 + 48 + 70 + 80 = 385$.
   • Calculate mean $\bar{x} = 385/10 = 38.5$.
   • Compute deviations $(x_i - \bar{x})^2$, sum them up, and calculate standard deviation $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{10}} \approx 23.5$.
8. 7 Marks
   bDescribe bivariate distribution. Write down the probability density function of bivariate normal distribution.
   How to answer

   • Define bivariate distribution as simultaneous distribution of two random variables $(X, Y)$.
   • Write PDF of bivariate normal distribution: $f(x,y) = \frac{1}{2\pi \sigma_X \sigma_Y \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)} Q(x,y) \right)$.
   • Specify $Q(x,y) = \left(\frac{x-\mu_X}{\sigma_X}\right)^2 - 2\rho\left(\frac{x-\mu_X}{\sigma_X}\right)\left(\frac{y-\mu_Y}{\sigma_Y}\right) + \left(\frac{y-\mu_Y}{\sigma_Y}\right)^2$.
   • Define all five parameters: $\mu_X, \mu_Y, \sigma_X, \sigma_Y, \rho$.
9. 7 Marks
   a

   Fit a second degree parabola to the following data:

   $x$	10	15	20	25	30	35	40
   $y$	11	13	16	20	27	34	41

   How to answer

   • State the equation of second degree parabola: $y = a + bx + cx^2$ (or use shifted variable $u = (x - 25)/5$).
   • Write normal equations using least squares method.
   • Construct table of values for $x, y, x^2, x^3, x^4, xy, x^2 y$ and their sums.
   • Formulate system of 3 linear equations.
   • Solve the system for $a, b, c$ and write the final fitted parabolic equation.
10. 7 Marks
    bWhat do you understand by Chi-square test of goodness of fit? Write condition for applying Chi-square test.
    How to answer

    • Define Chi-square goodness of fit test as a statistical test to check whether observed frequencies match theoretical expected frequencies.
    • State formula: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ with degrees of freedom $n - k - 1$.
    • State conditions for validity: total sample size $N$ must be reasonably large ($N \ge 50$).
    • Each theoretical cell frequency $E_i$ should not be less than 5 (combine cells if $<5$).
    • Observations must be independent and collected randomly.
11. 7 Marks
    aDefine Binomial distribution and obtain its mean and variance.
    How to answer

    • Define binomial distribution with PMF $P(X=x) = \binom{n}{x} p^x q^{n-x}$ for $x = 0, 1, \dots, n$, where $p + q = 1$.
    • Derive mean: $E[X] = \sum_{x=0}^n x \binom{n}{x} p^x q^{n-x} = np \sum_{x=1}^n \binom{n-1}{x-1} p^{x-1} q^{(n-1)-(x-1)} = np$.
    • Derive $E[X(X-1)] = n(n-1)p^2$.
    • Compute $E[X^2] = n(n-1)p^2 + np$.
    • Compute variance $V(X) = E[X^2] - (E[X])^2 = np(1-p) = npq$.
12. 7 Marks
    bDefine exponential distribution with parameter $\theta$ and obtain its mean, variance and moment generating function.
    How to answer

    • Define exponential distribution with parameter $\theta$: $f(x) = \theta e^{-\theta x}$ for $x \ge 0$.
    • Derive MGF: $M_X(t) = E[e^{tX}] = \int_0^\infty e^{tx} \theta e^{-\theta x} dx = \theta \int_0^\infty e^{-(\theta - t)x} dx = \frac{\theta}{\theta - t} = (1 - t/\theta)^{-1}$ for $t < \theta$.
    • Expand MGF or differentiate to get moments: $E[X] = M'_X(0) = 1/\theta$.
    • Find second moment: $E[X^2] = M''_X(0) = 2/\theta^2$.
    • Find variance: $V(X) = E[X^2] - (E[X])^2 = 2/\theta^2 - 1/\theta^2 = 1/\theta^2$.
13. 7 Marks
    aWhat do you mean by measure of skewness? Write tests of skewness.
    How to answer

    • Define skewness as the lack of symmetry in a frequency distribution.
    • Differentiate between positive skewness (tail to the right) and negative skewness (tail to the left).
    • List tests/indicators of skewness: mean $\ne$ median $\ne$ mode; quartiles not equidistant from median ($Q_3 - M \ne M - Q_1$); curve not bell-shaped.
    • Provide standard measures: Karl Pearson's coefficient $S_k = \frac{\text{Mean} - \text{Mode}}{\sigma}$ and Bowley's coefficient $S_k = \frac{Q_3 + Q_1 - 2M}{Q_3 - Q_1}$.
14. 7 Marks
    b

    Define Spearman's rank correlation coefficient and obtain rank correlation coefficient for the following data

    Rank in X	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
    Rank in Y	1	10	3	4	5	7	2	6	8	11	15	9	14	12	16	13

    How to answer

    • Define Spearman's rank correlation coefficient as a non-parametric measure of monotonic relationship between two ranked variables.
    • State formula $\rho = 1 - \frac{6\sum d_i^2}{n(n^2 - 1)}$, where $d_i = R_{Xi} - R_{Yi}$.
    • Calculate difference $d_i$ for each of the 16 pairs: (0, -8, 0, 0, 0, -1, 5, 2, 1, -1, -4, 3, -1, 2, -1, 3).
    • Compute squared differences $d_i^2$: (0, 64, 0, 0, 0, 1, 25, 4, 1, 1, 16, 9, 1, 4, 1, 9) and sum $\sum d_i^2 = 136$.
    • Substitute into formula with $n = 16$: $\rho = 1 - \frac{6 \times 136}{16(256 - 1)} = 1 - \frac{816}{4080} = 1 - 0.2 = 0.8$.
15. 7 Marks
    a

    Find the coefficient of correlation between X and Y

    X	1	2	3	4	5	6	7	8	9
    Y	12	11	13	15	14	17	16	19	18

    How to answer

    • State formula for Pearson's correlation coefficient $r$.
    • Calculate summary statistics: $\sum X = 45$, $\sum Y = 135$, $\bar{X} = 5$, $\bar{Y} = 15$.
    • Calculate $\sum X^2 = 285$, $\sum Y^2 = 2085$, $\sum XY = 731$.
    • Calculate numerator: $n\sum XY - (\sum X)(\sum Y) = 9(731) - (45)(135) = 6579 - 6075 = 504$.
    • Calculate denominator: $\sqrt{[9(285) - 45^2][9(2085) - 135^2]} = \sqrt{540 \times 540} = 540$.
    • Compute $r = 504 / 540 = 0.933$.
16. 7 Marks
    b

    Find the regression line of $y$ on $x$ for the following data:

    $x$	1	3	4	6	8	9	11	14
    $y$	1	2	4	4	5	7	8	9

    How to answer

    • State regression line equation of $y$ on $x$: $y - \bar{y} = b_{yx}(x - \bar{x})$.
    • Calculate sums: $n = 8$, $\sum x = 56$, $\sum y = 40$, $\bar{x} = 7$, $\bar{y} = 5$.
    • Calculate $\sum x^2 = 524$, $\sum xy = 364$.
    • Compute slope $b_{yx} = \frac{8(364) - (56)(40)}{8(524) - 56^2} = \frac{2912 - 2240}{4192 - 3136} = \frac{672}{1056} = \frac{7}{11} \approx 0.6364$.
    • Formulate equation: $y - 5 = 0.6364(x - 7) \implies y = 0.6364x + 0.545$.