AL-302 Introduction to Probability and Statistics - Dec 2025 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. 14 Marks
   A random variable x has the following probability function

   1. i)Determine k
   2. ii)Evaluate $P(x < 6)$ and $P(0 \le x \le 4)$.
   3. iii)If $P(x \le k) > \left(\frac{1}{2}\right)$ find the minimum value of k.
   4. iv)Determine the distribution function of k.
   5. v)Mean

   How to answer

   • Use sum of probabilities =1 to form equation in k
   • Solve 9k+3k^2+k=1 to get k=1/10
   • Compute cumulative sums for required probabilities
   • Write distribution function F(x) stepwise
   • Compute mean as sum x*P(x)
2. 7 Marks
   aThe density function of a random variable X is $f(x)=e^{-x}$ when $x \ge 0$. Find $E(X)$, $E(X^2)$ and variance of X.
   How to answer

   • Recognize exponential distribution with mean 1
   • Compute E(X)= integral x e^-x dx =1
   • Compute E(X^2)=2 via gamma integral
   • Variance = E(X^2)-[E(X)]^2 =1
3. 7 Marks
   bSuppose a continuous random variable X has the probability density function $f(x)= k (1-x^2)$ for $0 < x < 1$, and $f(x)=0$ otherwise. Find
   How to answer

   • Use normalization integral k*∫0^1 (1-x^2)dx=1 to find k=3/2
   • Compute mean ∫ x f(x)dx =3/8
   • Compute E(X^2) and variance = E(X^2)-mean^2
4. 7 Marks
   aWrite down merits and demerits of measure of central tendency. Find the median for the following data :

   How to answer

   • Define mean, median, mode and list merits/demerits
   • Form cumulative frequency table for wages data
   • Locate median class 4000-5000 via N/2
   • Apply median formula l+(N/2-cf)/f*h
5. 7 Marks
   bShow that sum of two independent random variables follows normal distribution.
   How to answer

   • Write mgf of normal variables
   • Use independence to multiply mgfs
   • Show product is mgf of normal form
   • Conclude sum is normally distributed
6. 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 P(|X-mean|>=k sigma)<=1/k^2
   • Write binomial pmf with parameters n,p
   • Take limit n→∞, p→0, np=lambda constant
   • Show pmf tends to e^-lambda lambda^x/x!
7. 7 Marks
   bDefine regression coefficient with its properties. If $X \sim N(\mu, \sigma^2)$, then show that $M_x(t) = e^{\mu t + \frac{1}{2}t^2\sigma^2}$.
   How to answer

   • Define regression coefficients b_xy, b_yx and properties
   • Write normal pdf and mgf definition
   • Evaluate integral M_X(t)=exp(mu t+sigma^2 t^2/2)
   • Conclude result
8. 7 Marks
   aFit a second degree parabola to the following data :

   How to answer

   • Assume y=a+bx+cx^2 and form normal equations
   • Compute sums Σx, Σx^2, Σx^3, Σx^4, Σy, Σxy, Σx^2y
   • Solve 3 equations for a,b,c
   • Write fitted parabola
9. 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 statistic
   • State null hypothesis of observed vs expected fit
   • List conditions: N large, expected freq >=5, etc.
   • Give decision rule using chi-square table
10. 7 Marks
    aDefine Binomial distribution and obtain its mean and variance.
    How to answer

    • Define binomial pmf C(n,x)p^x q^{n-x}
    • Derive mean = np using expectation sum
    • Derive variance = npq
11. 7 Marks
    bDefine exponential distribution with parameter $\theta$ and obtain its mean, variance and moment generating function.
    How to answer

    • Define exponential pdf with parameter theta
    • Compute mean =1/theta or theta by parametrization
    • Compute variance via second moment
    • Derive mgf M(t)=theta/(theta-t)
12. 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

    1. i)at least one
    2. ii)at most one.
    How to answer

    • Identify Poisson with lambda=1.6
    • Compute P(X>=1)=1-e^-lambda
    • Compute P(X<=1)=e^-lambda(1+lambda)
    • Evaluate numerically
13. 7 Marks
    bDefine correlation. Explain types of correlation and method of studying correlation.
    How to answer

    • Define correlation as measure of linear relationship
    • Explain positive, negative, zero and linear/non-linear types
    • Describe scatter diagram, Karl Pearson, Spearman methods
    • Give merits of each method
14. 7 Marks
    aFind Karl Pearson's coefficient of correlation from the following data -

    How to answer

    • Compute means of x and y
    • Compute Σ(x-xbar)(y-ybar), Σ(x-xbar)^2, Σ(y-ybar)^2
    • Apply r = covariance/(sx sy)
    • Interpret value
15. 7 Marks
    bFind the regression line of y on x for the following data :

    How to answer

    • Compute means xbar, ybar
    • Compute regression coefficient b_yx = Cov/var(x)
    • Form equation y-ybar = b_yx (x-xbar)
    • Simplify to y = a+bx form