University Of Nottingham School Of Applied Mathematic 856267

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Analyze a set of probability and statistical problems involving Bayesian updating, density functions, Poisson and binomial distributions, and probability modeling of dart game scores. The assignments include calculating posterior probabilities, density normalization, cumulative distribution functions, expected waiting times, event probabilities, and evaluating the fairness of a game based on statistical evidence.

Paper For Above instruction

In this paper, we explore various fundamental and advanced concepts in probability and statistics, applying them to real-world scenarios and theoretical models. We begin with Bayesian inference to identify causes of an event, proceed to density and distribution functions, and then analyze timing and counting processes governed by Poisson and binomial models. The final sections involve scoring probabilities within a game context and evaluating whether the game is statistically fair.

Question 1: Bayesian Analysis and Probability Density Functions

(a) Causes of an Explosion at a Construction Site

The problem provides four potential causes for an explosion: static electricity (A), malfunctioning of equipment (B), carelessness (C), and sabotage (D). Each cause has assigned prior probabilities and conditional probabilities of causing an explosion. The goal is to compute the posterior probabilities for each cause given an explosion and identify the most probable cause.

Before the explosion, prior probabilities are:

  • P(A) = 0.2
  • P(B) = 0.45
  • P(C) = 0.25
  • P(D) = 0.1

The conditional probabilities of an explosion given each cause are:

  • P(E | A) = 0.35
  • P(E | B) = 0.2
  • P(E | C) = 0.5
  • P(E | D) = 0.6

Applying Bayes' theorem, the posterior probability for each cause is calculated as:

P(A | E) = (P(E | A) * P(A)) / P(E)

Similarly for B, C, D. The total probability of an explosion, P(E), is found by summing over all causes:

P(E) = P(E | A) P(A) + P(E | B) P(B) + P(E | C) P(C) + P(E | D) P(D)

Substituting values:

P(E) = (0.35)(0.2) + (0.2)(0.45) + (0.5)(0.25) + (0.6)(0.1) = 0.07 + 0.09 + 0.125 + 0.06 = 0.345

Calculating each posterior:

P(A | E) = (0.35)(0.2) / 0.345 ≈ 0.07 / 0.345 ≈ 0.203

P(B | E) = (0.2)(0.45) / 0.345 ≈ 0.09 / 0.345 ≈ 0.261

P(C | E) = (0.5)(0.25) / 0.345 ≈ 0.125 / 0.345 ≈ 0.362

P(D | E) = (0.6)(0.1) / 0.345 ≈ 0.06 / 0.345 ≈ 0.174

Most likely cause is C (carelessness) with approximately 36.2% probability.

(b) Density Functions and Probability Calculations

The probability density function (PDF) of the measurement of sulphur dioxide concentration is given by a specific form. To find the normalization constant, integrate the PDF over its support and set the integral to 1.

Suppose the density is f(x) = k * g(x) over support [a, b], then:

k = 1 / ∫[a, b] g(x) dx

Similarly, the cumulative distribution function (CDF) is obtained by integrating the PDF from the lower bound up to x:

F(x) = ∫[a, x] f(t) dt

If a second pollutant's content depends on the sulphur dioxide measurement, its PDF can be derived using transformation of variables. The probability of the measurement falling within a certain range is computed by evaluating this integral or using the CDF.

For example, if the range of interest is [x1, x2], then: P(x1 ≤ X ≤ x2) = F(x2) – F(x1).

Numerical calculations depend on the explicit form of g(x) and bounds, which should be integrated accordingly to find the exact probabilities.

Question 2: Poisson and Binomial Models in Timing and Counting

(a) Inter-Arrival Times in a Poisson Process

Jobs are sent to a printer at an average rate λ = 4 jobs/hour. The Poisson process describes the number of jobs arriving in a given interval, and the inter-arrival times follow an exponential distribution with parameter λ.

  • Expected time between jobs: E(T) = 1 / λ = 1 / 4 hours = 15 minutes.
  • The probability that a job arrives within 5 minutes is P(T ≤ 5) = 1 – exp(–λ 5/60) ≈ 1 – exp(–4 5/60) ≈ 1 – exp(–1/3) ≈ 0.2835.
  • Maximum expected waiting time for the first arrival with probability 0.9 is derived from the exponential distribution's inverse CDF: T = –(1/λ) ln(1 – 0.9) ≈ – (1/4) ln(0.1) ≈ 0.5753 hours ≈ 34.52 minutes.

For the second case, given that the first job was at 9:30 a.m., the time until the second is again exponential with the same rate, independent of the first. The probability that the second arrives before 9:45 a.m. (15 minutes after the first) is:

P(T ≤ 15) = 1 – exp(–4 * 15/60) = 1 – exp(–1) ≈ 0.6321.

The probability that it arrives after 9:45 is 1 – 0.6321 = 0.3679. This relates closely to the memoryless property of the exponential and the Poisson process, where the waiting time for the next event does not depend on the history.

(b) Binomial Model for Seed Germination

The plant's germination process, with each seed independently having a 90% chance to germinate, is modeled as a binomial random variable X ~ Bin(n=60, p=0.9).

Since p is constant for each trial and trials are independent, the binomial distribution applies naturally.

  • The probability that at least 55 seeds germinate is:

P(X ≥ 55) = 1 – P(X ≤ 54). The exact calculation is cumbersome, but it can be computed using binomial tables or software. Alternatively, normal approximation can be employed with continuity correction.

Mean and standard deviation are:

μ = np = 60 * 0.9 = 54

σ = √(np(1–p)) = √(600.90.1) ≈ √5.4 ≈ 2.32

Using the normal approximation with continuity correction (X ≥ 55.5):

P(X ≥ 55) ≈ P(Z ≥ (54.5 – 54)/2.32) ≈ P(Z ≥ 0.216) ≈ 0.4157.

The approximation's suitability is justified because np and n(1–p) are sufficiently large, making the normal approximation reliable in this context.

Question 3: Geometric Probabilities and Game Fairness

(a) Dartboard Probabilities

The dartboard has three concentric circles with radii √ , 1, and √ meters. The density function for the distance, R, from the center is proportional to r over the support.

Under uniform distribution over the area, the probability of a dart falling within a certain radius is proportional to the area of the corresponding circle (since area = πr²):

  • Probability within the smallest circle (radius √):

P(r ≤ √) = (Area of circle with radius √) / (Area of entire board)

= π(√)² / π(√)² = 1, for the inner circle.

Similarly, between circles with radii √ and 1, the probability is:

P(√

In general, the probabilities are proportional to the difference of areas of the respective annular regions divided by the total area, assuming uniform distribution.

Specific calculations depend on the explicit radii and density assumptions provided, but the fundamental approach involves ratios of areas.

(b) Evaluating Fairness of the Game

The game involves five attempts, each scoring points based on where the dart lands: 5, 3, 2, or 0 points, depending on the ring, at a cost of $15 per game. The total score determines the payout.

Assuming the probabilities of landing in each scoring zone are known, the expected total score can be calculated as:

E[Score] = 5 P(inner circle) + 3 P(second ring) + 2 P(third ring) + 0 P(outside).

Using the probabilities derived in (a), for each attempt, the expected score is this value multiplied by five (because there are five shots). The fairness of the game can then be assessed by comparing the expected payout with the cost of the game ($15), and analyzing the variance and probability of exceeding this cost.

If the expected score (and payout) exceeds the cost, the game favors the player; otherwise, it favors the house. Statistical evidence, such as the expected value and variance, would support the conclusion about fairness.

In practice, detailed probabilities and payout structures must be specified, but under the assumptions of uniform distribution and known scoring zones, the statistical analysis demonstrates whether the game is fair or biased.

Conclusion

This study demonstrates the application of Bayesian inference in fault diagnosis, density and cumulative functions in measurement analysis, timing and count modeling with Poisson and binomial distributions, and game theory evaluation through probability modeling. By integrating these probabilistic methods, we gain insights into complex systems and decision-making scenarios, underlining the importance of statistical reasoning in engineering and applied mathematics.

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