AL-302 Introduction to Probability and Statistics - Dec 2024 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
   aIf $X$ is a continuous random variable and $K$ is a constant. Then prove that $\text{Var}(X + k) = \text{Var}(X)$ and $\text{Var}(kX) = k^2 \text{Var}(X)$.
   How to answer

   • State definition of variance: Var(Y) = E[(Y - E[Y])^2]
   • Evaluate E[X + k] = E[X] + k and substitute into variance formula to show Var(X + k) = Var(X)
   • Evaluate E[kX] = kE[X] and factor out k^2 from E[(kX - kE[X])^2] to show Var(kX) = k^2 Var(X)
2. 7 Marks
   bState and Prove Chebyshev's inequality.
   How to answer

   • State Chebyshev's inequality statement
   • Express variance integral and split into three regions
   • Drop non-negative middle integral and bound lower/upper integrals
   • Conclude with standard inequality form
3. 7 Marks
   aA random variable $X$ is defined as the sum of the numbers on the faces when two dice are thrown. Find the mean of $X$.
   How to answer

   • Identify sample space of two dice (36 outcomes)
   • Determine PMF for X ranging from 2 to 12
   • Apply formula E[X] = sum(x * P(x))
   • Calculate final result E[X] = 7
4. 7 Marks
   bExplain Exponential and Gamma densities with properties.
   How to answer

   • Define Exponential PDF f(x) = lambda * e^(-lambda*x) for x >= 0
   • List key properties of Exponential distribution (memoryless property, mean = 1/lambda, variance = 1/lambda^2)
   • Define Gamma PDF f(x) = (beta^alpha / Gamma(alpha)) * x^(alpha-1) * e^(-beta*x) for x > 0
   • List key properties of Gamma distribution (mean = alpha/beta, variance = alpha/beta^2, relationship to exponential)
5. 14 Marks
   The probability density function of a random variable $X$ is $$f(x) = \begin{cases} \frac{\sin x}{2}, & \text{for } 0 \le x \le \pi \\ 0, & \text{elsewhere} \end{cases}$$ Find the mean, mode and median of a distribution and also find the probability between $0$ and $\frac{\pi}{2}$.
   How to answer

   • Compute Mean: E[X] = int_0^pi x*(sin x / 2) dx = pi/2
   • Find Mode: maximize f(x) = sin(x)/2 -> f'(x) = cos(x)/2 = 0 => x = pi/2
   • Find Median M: int_0^M (sin x / 2) dx = 1/2 => [-cos x / 2]_0^M = 1/2 => cos M = 0 => M = pi/2
   • Compute P(0 <= X <= pi/2) = int_0^(pi/2) (sin x / 2) dx = 1/2
6. 7 Marks
   aDefine Bivariate distribution and explain their properties.
   How to answer

   • Define joint probability function (discrete: PMF, continuous: PDF) for two random variables X and Y
   • State property: f(x, y) >= 0 and double integral/sum equals 1
   • Explain marginal probability distributions of X and Y
   • Explain conditional probability distributions and independence
7. 7 Marks
   bIf first box contains 2 black, 3 red, 1 white balls; Second box contains 1 black, 1 red, 2 white balls; Third box contains 5 black, 3 red, 4 white balls. Of these a box is selected at random. From it a red ball is randomly drawn. If the ball is red, find the probability that it is from second box.
   How to answer

   • Define events: B1, B2, B3 for selecting boxes, R for drawing a red ball
   • Calculate prior probabilities: P(B1) = P(B2) = P(B3) = 1/3
   • Determine conditional probabilities: P(R|B1) = 3/6 = 1/2, P(R|B2) = 1/4, P(R|B3) = 3/12 = 1/4
   • Compute total probability P(R) = (1/3)*(1/2 + 1/4 + 1/4) = 1/3
   • Apply Bayes' theorem: P(B2|R) = ( (1/3)*(1/4) ) / (1/3) = 1/4
8. 7 Marks
   aThe values of $\mu_1$, $\mu_2$, $\mu_3$ and $\mu_4$ are $0$, $9.2$, $3.6$ and $1.22$ respectively. Find out the Skewness and Kurtosis of the distribution.
   How to answer

   • State formula for coefficient of skewness: beta_1 = mu_3^2 / mu_2^3
   • Compute beta_1 = (3.6)^2 / (9.2)^3 = 12.96 / 778.688 \approx 0.0166 (or gamma_1 = sqrt(beta_1) \approx 0.129)
   • State formula for kurtosis: beta_2 = mu_4 / mu_2^2
   • Compute beta_2 = 1.22 / (9.2)^2 = 1.22 / 84.64 \approx 0.0144
9. 7 Marks
   bThe mean and standard deviation of a normal variable are $8$ and $4$ respectively. Find: $P(5 \le X \le 10)$ and $P(X \ge 5)$
   How to answer

   • Convert X to standard normal Z: Z = (X - 8) / 4
   • For X = 5: Z1 = (5 - 8)/4 = -0.75; for X = 10: Z2 = (10 - 8)/4 = 0.5
   • Calculate P(5 <= X <= 10) = P(-0.75 <= Z <= 0.5) = \Phi(0.5) - \Phi(-0.75)
   • Calculate P(X >= 5) = P(Z >= -0.75) = 1 - \Phi(-0.75) = \Phi(0.75)
10. 7 Marks
    aAverage number of accidents on any day on a national highway is $1.6$. Determine the probability that the number of accidents are i) at least one ii) at most one
    How to answer

    • Model number of accidents using Poisson distribution with lambda = 1.6
    • Calculate P(X = 0) = e^(-1.6) \approx 0.2019
    • i) P(X >= 1) = 1 - P(X = 0) = 1 - e^(-1.6) \approx 0.7981
    • Calculate P(X = 1) = 1.6 * e^(-1.6) \approx 0.3230
    • ii) P(X <= 1) = P(X = 0) + P(X = 1) = e^(-1.6) * (1 + 1.6) = 2.6 * e^(-1.6) \approx 0.5249
11. 7 Marks
    bDefine correlation. Explain types of correlation and methods of studying correlation.
    How to answer

    • Define correlation as the statistical measure of association between two variables
    • Explain types: Positive/Negative, Linear/Non-linear, Simple/Partial/Multiple
    • Describe methods: Scatter diagram method, Karl Pearson's coefficient of correlation, Spearman's rank correlation method, Method of concurrent deviations
12. 7 Marks
    a

    Fit a Straight line for the following data.

    X	0	5	10	15	20	25
    Y	12	15	17	22	24	30

    How to answer

    • Assume straight line equation: Y = a + bX, n = 6
    • Calculate summations: sum(X) = 75, sum(Y) = 120, sum(X^2) = 1375, sum(XY) = 1795
    • Write normal equations: 6a + 75b = 120 and 75a + 1375b = 1795
    • Solve for a and b: b = 0.7086, a = 11.143
    • State final fitted line: Y = 11.14 + 0.71X
13. 7 Marks
    bA sample of $400$ items is taken from a population whose standard deviation is $10$. The mean of the sample is $40$. Test whether the sample has come from a population with mean $38$. Also calculate $95\%$ confidence interval for the population.
    How to answer

    • Formulate hypotheses: H0: mu = 38 vs H1: mu != 38
    • Calculate standard error: SE = sigma / sqrt(n) = 10 / 20 = 0.5
    • Compute test statistic: Z = (40 - 38) / 0.5 = 4.0
    • Compare with critical value at 5% level (1.96): Since |Z| = 4.0 > 1.96, reject H0 (sample did not come from this population)
    • Compute 95% CI: 40 \pm 1.96 * 0.5 = 40 \pm 0.98 = [39.02, 40.98]
14. 7 Marks
    aDefine Chi-Square test of goodness of fit. Explain the conditions for Chi-Square test.
    How to answer

    • Define Chi-Square goodness of fit test with formula: chi^2 = sum((O - E)^2 / E)
    • State degrees of freedom (df = n - 1 - k)
    • Explain conditions: Total frequency N should be reasonably large (N >= 50)
    • No theoretical frequency should be less than 5
    • Observations must be independent and sample drawn randomly
15. 7 Marks
    b

    Calculate coefficient of correlation from the following data.

    X	12	9	8	10	11	13	7
    Y	14	8	6	9	11	12	3

    How to answer

    • Set up table with N = 7: X, Y, X^2, Y^2, XY
    • Compute sums: sum(X) = 70, sum(Y) = 63, sum(X^2) = 728, sum(Y^2) = 651, sum(XY) = 686
    • Means: bar(X) = 10, bar(Y) = 9
    • Compute deviations or use direct formula: r = 56 / sqrt(28 * 84) = 56 / sqrt(2352) = 56 / 48.497
    • Conclude r \approx +0.955 (high positive correlation)