Engineering Math - Correlation and Probability

• These notes cover topics from Engineering Mathematics - IV.

Correlation Coefficient

• Calculate the correlation coefficient between variables x and y using the provided data points.
• Data points include x values: 18, 20, 34, 52, 12
• Corresponding y values: 39, 23, 35, 18, 46.

Random Variables and Probability

• A random variable x has a probability function.
• The probabilities are defined as follows: P(0) = 0, P(1) = C, P(2) = 2C, P(3) = 2C, P(4) = 3C, P(5) = 2C.
• Determine i) C, ii) P(x<3), iii) E(X), and iv) V(X).

Hypothesis Testing for Bulb Lifetimes

• The mean lifetime of a sample of 25 bulbs is 1550 hours, with a standard deviation of 120 hours.
• The bulb manufacturer claims an average life of 1600 hours.
• Test if the claim is acceptable at a 5% Level of Significance (LOS).

Vector Calculus - Solenoidal and Irrotational Fields

• A vector field F is defined as F = (x + 2y + 4z)i + (2x – 3y - z)j + (4x - y + 2z)k.
• Prove that F is both solenoidal and irrotational.

Linear Regression

• Fit a straight line to the given dataset consisting of x and y values.
• The x values are 1, 2, 3, 4, 5.
• The corresponding y values are 25, 28, 33, 39, 46.

Work Done by a Force Field

• Calculate the work done in moving a particle in the force field F = (3x² + 6y)i – 14yzj + 20xz²k.
• The particle moves along the curve x = t, y = t, z = t from point (0,0,0) to (1,1,1).

Laurent Series Expansion

• Find all possible Laurent series expansions of the function f(z) = (z-1) / ((z+1)(z-3)) about z = 0.
• Indicate the region of convergence for each series.

Regression Lines

• Given two regression lines for a sample: x + 6y = 6 and 3x + 2y = 10.
• Determine the values of the means x̄ and ȳ.
• Calculate the correlation coefficient r.
• Estimate the value of y when x = 12.

Green's Theorem

• Use Green's theorem to evaluate the line integral ∫(3x² - 8y²) dx + (4y - 6xy) dy.
• C is the boundary of the region enclosed by the lines x = 0, y = 0, and x + y = 1.

Hypothesis Testing for Drug Effectiveness

• A drug's effectiveness in curing the common cold is tested on 500 people.
• 300 people receive the drug, while 200 receive a placebo (sugar pills).
• The results are recorded in a table.
• Test the hypothesis that the drug is effective in curing colds using a Chi-squared test (χ²) with a 5% Level of Significance (LOS).
• The table shows:
  • Drug: 200 helped, 40 harmed, 60 no effect out of 300 total.
  • Sugar pills: 120 helped, 30 harmed, 50 no effect out of 200 total.
  • Totals: 320 helped, 70 harmed, 110 no effect out of 500 total.

Probability Density Function

• X is a continuous random variable with probability density function f(x) = k(x - x²) for 0 ≤ x ≤ 1.
• Find the value of k, the mean, and the variance.

Variance Hypothesis Testing

• Results were obtained from two samples drawn from two different populations, A and B.
• For group A: Sample Size = 25, Sample SD = 4.
• For group B: Sample Size = 17, Sample SD = 3.
• Test the hypothesis that the variance of A is less than or equal to the variance of B, given F(0.05) = 2.24 (degrees of freedom 24 and 16).

Conservative Vector Fields

• Show that the vector field F = (6xy + z³)i + (3x² – z)j + (3xz² – y)k is conservative.
• Find a scalar potential such that F = ∇Ø.
• Calculate the work done in displacing a particle from (1,2,0) to (3,3,2).

Moment Generating Function (MGF)

• If X denotes the outcome when a fair die is tossed, find the Moment Generating Function (MGF) of X about the origin.
• Determine the mean of X.

Stoke's Theorem

• Use Stoke's Theorem to evaluate the integral of F ⋅ dr, where F = (x² - y²)i + 2xyj.
• C is the boundary of the region defined by x = 0, y = 0, x = 4, and y = 2.

Complex Integration

• Evaluate the integral ∮ (z²+3)/((z-1)(z-2)) dz, where c is defined by:
  • (i) |z - 1| = 1
  • (ii) |z + 1| = 1

Bayes' Theorem Application

• Three factories, A, B, and C, produce 30%, 50%, and 20% of the total production of an item, respectively.
• Defective rates: Factory A (80%), Factory B (50%), and Factory C (10%).
• An item is chosen at random and found to be defective.
• Use Bayes' theorem to find the probability that this item was produced by factory A.

Gauss Divergence Theorem

• Use the Gauss Divergence Theorem to evaluate the surface integral of F ⋅ n ds.
• The vector field F is given by F = x³i + y³j + z³k.
• S is the surface of the sphere defined by x² + y² + z² = 1.

Intelligence Test Statistics

• In an intelligence test administered to 1000 students, the average score was 42, and the standard deviation was 24.
• Determine the number of students:
  • (i) Exceeding a score of 50.
  • (ii) Scoring between 30 and 54.
  • (iii) Scoring less than 30.