AL-302 Introduction to Probability and Statistics - Jun 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 continuous random variable. Also show that $$E(X_1 X_2 \cdots X_n) = E(X_1) E(X_2) \cdots E(X_n)$$
   How to answer

   • Define continuous random variable with support and probability density function (PDF).
   • State condition of independence among random variables $X_1, X_2, \dots, X_n$.
   • Write the joint PDF as product of marginal PDFs: $f(x_1, x_2, \dots, x_n) = \prod f_i(x_i)$.
   • Express $E(X_1 X_2 \dots X_n)$ as multiple integral.
   • Factorize the integral into product of individual expectations.
2. 7 Marks
   bDefine non-central and central moments. Also, show that $$V(2X + 3) = 4V(X)$$
   How to answer

   • Define $r$-th raw/non-central moment $\mu_r' = E[(X-a)^r]$ (or about origin: $E[X^r]$).
   • Define $r$-th central moment $\mu_r = E[(X-\mu)^r]$.
   • Use definition of variance $V(Y) = E[(Y - E[Y])^2]$.
   • Substitute $Y = 2X + 3$, find $E[Y] = 2E[X] + 3$.
   • Simplify $V(2X + 3) = E[(2X - 2E[X])^2] = 4E[(X-E[X])^2] = 4V(X)$.
3. 7 Marks
   aShow that Poisson distribution is a limiting case of binomial distribution under the case $n \to \infty$, $p \to 0$ and $np = m$.
   How to answer

   • State PMF of Binomial distribution $P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$.
   • Substitute $p = m/n$ and $q = 1 - m/n$.
   • Expand $\binom{n}{x} = \frac{n(n-1)\cdots(n-x+1)}{x!}$.
   • Take limit as $n \to \infty$ for each term.
   • Use standard limit $\lim_{n \to \infty} (1 - m/n)^n = e^{-m}$ to obtain $\frac{e^{-m} m^x}{x!}$.
4. 7 Marks
   bDefine gamma distribution. If X follows exponential distribution with parameter $\theta$, then obtain its mean and variance.
   How to answer

   • Define PDF of Gamma distribution with parameters $\alpha$ and $\beta$.
   • Write PDF of exponential distribution: $f(x) = \theta e^{-\theta x}$ (or $\frac{1}{\theta}e^{-x/\theta}$) for $x \ge 0$.
   • Derive mean: $E[X] = \int_0^\infty x f(x) dx$.
   • Derive $E[X^2] = \int_0^\infty x^2 f(x) dx$.
   • Compute variance: $V(X) = E[X^2] - (E[X])^2$.
5. 7 Marks
   aIf $X \sim N(\mu, \sigma^2)$, then show that $\varphi_x(t) = e^{i\mu t - \frac{1}{2}t^2 \sigma^2}$
   How to answer

   • Define characteristic function $\varphi_X(t) = E[e^{itX}] = \int_{-\infty}^{\infty} e^{itx} f(x) dx$.
   • Transform variable using $Z = \frac{X - \mu}{\sigma}$, so $X = \mu + \sigma Z$.
   • Express $\varphi_X(t) = e^{it\mu} \varphi_Z(\sigma t)$.
   • Evaluate standard normal integral $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{itz - z^2/2} dz$ by completing the square.
   • Conclude $\varphi_X(t) = e^{i\mu t - \frac{1}{2}t^2 \sigma^2}$.
6. 7 Marks
   bA large number of measurement is normally distributed with a mean $65.5\text{ cm}$ and S.D. of $6.2\text{ cm}$. Find the percentage of measurement that fall between $54.8\text{ cm}$ and $68.8\text{ cm}$.
   How to answer

   • Calculate standard normal score $z_1 = \frac{54.8 - 65.5}{6.2} = \frac{-10.7}{6.2} \approx -1.73$.
   • Calculate standard normal score $z_2 = \frac{68.8 - 65.5}{6.2} = \frac{3.3}{6.2} \approx 0.53$.
   • Find area $P(-1.73 < Z < 0.53) = \Phi(0.53) - \Phi(-1.73)$.
   • Obtain area from standard normal tables.
   • Multiply by 100 to state the required percentage.
7. 7 Marks
   a

   Write down merits and demerits of measure of central tendency. Find the median for the following data;

   Wages	2000-3000	3000-4000	4000-5000	5000-6000	6000-7000
   No. of workers	3	5	20	10	5

   How to answer

   • List merits and demerits of main measures of central tendency (Mean, Median, Mode).
   • Form cumulative frequency table ($N = 43$, $N/2 = 21.5$).
   • Identify median class (4000-5000).
   • Apply median formula: $\text{Median} = L + \left(\frac{N/2 - cf}{f}\right) \times h$.
   • Substitute values: $L=4000, cf=8, f=20, h=1000$ to obtain median.
8. 7 Marks
   bShow that sum of two independent random variables follows normal distribution.
   How to answer

   • State MGF or characteristic function of two independent normal random variables $X_1$ and $X_2$.
   • Express MGF of sum $Y = X_1 + X_2$ as $M_Y(t) = M_{X_1}(t) \cdot M_{X_2}(t)$.
   • Multiply the two exponentials: $e^{(\mu_1 + \mu_2)t + \frac{1}{2}(\sigma_1^2 + \sigma_2^2)t^2}$.
   • Recognize this as the MGF of a normal distribution with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$.
   • By uniqueness theorem of MGF, conclude that $Y$ follows normal distribution.
9. 7 Marks
   a

   Using method of Least squares, find the curve $y = ax + bx^2$ that best fit the following data:

   $x$	1	2	3	4	5
   $y$	1.8	5.1	8.9	14.1	19.8

   How to answer

   • Divide curve by $x$: $y/x = a + bx$ or write normal equations for $y = ax + bx^2$ directly: $\sum xy = a\sum x^2 + b\sum x^3$ and $\sum x^2y = a\sum x^3 + b\sum x^4$.
   • Form computation table for $x, y, x^2, x^3, x^4, xy, x^2y$.
   • Compute summations: $\sum x^2, \sum x^3, \sum x^4, \sum xy, \sum x^2y$.
   • Formulate the system of two linear equations in $a$ and $b$.
   • Solve for parameters $a$ and $b$ and write the fitted equation.
10. 7 Marks
    bOut of 8000 graduates in a town, 800 are females, out of 1600 graduate employees, 120 are females. Use $\chi^2$ test to determine if any distinction is made in appointment on the basis of sex. The value of $\chi^2$ for 1 degree of freedom at 5% level is 3.841.
    How to answer

    • Set up null hypothesis $H_0$: There is no distinction made in appointment on the basis of sex.
    • Formulate a $2 \times 2$ contingency table: Employed/Unemployed vs Male/Female.
    • Calculate expected frequencies $E_{ij}$ for all cells.
    • Compute $\chi^2 = \sum \frac{(O - E)^2}{E}$.
    • Compare calculated $\chi^2$ with critical value 3.841 and state inference.
11. 7 Marks
    aDefine Poisson distribution and obtain its mean and variance.
    How to answer

    • Define Poisson PMF: $P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$, $x = 0, 1, 2, \dots$.
    • Derive mean $E[X] = \sum_{x=0}^{\infty} x P(X = x) = \lambda$.
    • Derive $E[X(X-1)] = \sum_{x=0}^{\infty} x(x-1) P(X = x) = \lambda^2$.
    • Find $E[X^2] = \lambda^2 + \lambda$.
    • Compute variance $V(X) = E[X^2] - (E[X])^2 = \lambda$.
12. 7 Marks
    bDefine gamma distribution with parameter $\lambda$ and obtain its mean, variance, and characteristic function.
    How to answer

    • Define PDF of Gamma variate with parameter $\lambda$: $f(x) = \frac{e^{-x} x^{\lambda - 1}}{\Gamma(\lambda)}$, $x > 0$.
    • Derive mean $E[X] = \int_0^\infty x f(x) dx = \frac{\Gamma(\lambda + 1)}{\Gamma(\lambda)} = \lambda$.
    • Derive $E[X^2] = \frac{\Gamma(\lambda + 2)}{\Gamma(\lambda)} = \lambda(\lambda + 1)$ and variance $V(X) = \lambda$.
    • Evaluate characteristic function $\varphi_X(t) = \int_0^\infty e^{itx} f(x) dx$.
    • Simplify integral to obtain $\varphi_X(t) = (1 - it)^{-\lambda}$.
13. 7 Marks
    aWhat do you mean by measure of kurtosis? Obtain $\beta_1$ and $\beta_2$ for the following function: $$f(x) = y_0 x(2-x); \quad 0 \le x \le 2$$
    How to answer

    • Define kurtosis (peakedness of distribution) and define $\beta_1 = \frac{\mu_3^2}{\mu_2^3}$ and $\beta_2 = \frac{\mu_4}{\mu_2^2}$.
    • Determine $y_0$ using normalization condition $\int_0^2 f(x) dx = 1$.
    • Calculate mean $\mu = E[X] = \int_0^2 x f(x) dx$ (note symmetry about $x=1$).
    • Calculate central moments $\mu_2, \mu_3$ (which is 0 by symmetry), and $\mu_4$.
    • Compute $\beta_1 = 0$ and calculate $\beta_2$.
14. 7 Marks
    b

    Define Karl Pearson's correlation coefficient and obtain the correlation coefficient for the following data:

    X	65	66	67	67	68	69	70	72
    Y	67	68	65	68	72	72	69	71

    How to answer

    • Define Karl Pearson's correlation coefficient $r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$.
    • Construct table of $X, Y, u = X - \bar{X}, v = Y - \bar{Y}, u^2, v^2, uv$ (or use raw data).
    • Calculate means $\bar{X}$ and $\bar{Y}$.
    • Calculate $\sum (X-\bar{X})^2$, $\sum (Y-\bar{Y})^2$, and $\sum (X-\bar{X})(Y-\bar{Y})$.
    • Substitute values into Pearson formula to find $r$.
15. 7 Marks
    a

    Find the rank correlation coefficient to the following data:

    $x$	65	63	67	64	68	62	70	66	68	67	69	71
    $y$	68	66	68	65	69	66	68	65	71	67	68	70

    How to answer

    • Assign ranks to values of $x$ and $y$, resolving ties by averaging ranks.
    • Compute rank difference $d = R_x - R_y$ and $d^2$ for each observation.
    • Identify tied values in $x$ and $y$ and calculate tie correction factors $\frac{m(m^2-1)}{12}$.
    • Compute $\sum d^2$ and total corrected sum.
    • Apply Spearman rank correlation formula with tied ranks correction to find $r_s$.
16. 7 Marks
    bDefine binomial distribution and derive its moment generating function. Hence, obtain its mean and variance.
    How to answer

    • Define Binomial distribution PMF: $P(X=x) = \binom{n}{x} p^x q^{n-x}$, $x=0,1,\dots,n$.
    • Derive MGF: $M_X(t) = E[e^{tX}] = \sum_{x=0}^n e^{tx} \binom{n}{x} p^x q^{n-x} = (q + pe^t)^n$.
    • Find first derivative $M_X'(t)|_{t=0}$ to obtain Mean = $np$.
    • Find second derivative $M_X''(t)|_{t=0} = E[X^2] = n(n-1)p^2 + np$.
    • Compute Variance = $E[X^2] - (E[X])^2 = np(1-p) = npq$.