Assignment 1: Probability

Assignment 1: Probability
Instructions
Complete the following problem set showing your work. Problems may be worked out "by hand" or in "python" or with the
assistance of other analytical software (e.g., Mathematica, MatLab). You may use chatGPT to assist in coding.
Solutions must be type written (e.g., in Jupyter, markdown, or latex). Upload PDF solution by question to crowdmark (link will be
emailed to you) by 11:59pm on the Saturday of the corresponding week (see syllabus).
All problems are equally weighted within an assignment. Students in 468 may or may not choose to attempt the challenge
question for a bonus pts. Students in 795 are required to complete the challenge question.
Problem Set
1. Infectious Disease Testing
A new infectious disease has emerged, and a diagnostic test has been developed to identify individuals who are infected. The test
is not perfect and can yield both false positives and false negatives. You are given the following information:
The prevalence of the disease in the population is 5%, meaning that 5% of individuals are infected.
The sensitivity of the test is 90%, which means it correctly identifies 90% of infected individuals as positive.
The specificity of the test is 85%, which means it correctly identifies 85% of uninfected individuals as negative.
Part A. Given that an individual tests positive, what is the probability that they are truly infected? Use Bayes' theorem to calculate
this quantity which is known as the positive predictive value (PPV). Is this a high or low PPV?
Part B. Given that an individual tests negative, what is the probability that they are truly uninfected? Use Bayes' theorem to
calculate the negative predictive value (NPV). Is this a high or low NPV?
Part C. Discuss the impact of test sensitivity and specificity on the accuracy of diagnostic tests, especially in the context of
infectious diseases.
Part D. Suppose that individuals are tested twice. Assuming that the accuracy of the tests is independent, what is the probability
that an individual is infected given two negative test results? Discuss possible pros and cons of a multiple-testing design
2. Genetic Inheritance and Probability
In genetics, the principles of probability are often used to understand the inheritance of traits. Consider a specific genetic trait
determined by a single gene with two alleles: dominant (D) and recessive (d). In a population, 40% of individuals are homozygous
dominant (DD), 30% are heterozygous (Dd), and 30% are homozygous recessive (dd) for this trait.
Part A. What is the probability that a randomly selected individual exhibits the dominant trait?
Part B. Given that two individuals heterozygous for this trait (Dd) mate and have offspring, what is the probability that their child
will express the dominant trait)?
Part C. Consider a different co-dominant genetic trait determined by a separate (unlinked) gene. In the focal population, 60% of
individuals express the mutant phenotype, 30% express the heterozygous phenotype, and 10% express the ancestral phenotype.
What is the probability that a randomly selected individual is heterozygous for the first trait (Dd) and expresses the mutant
phenotype (MM) for the second trait?
Part D. Discuss the importance of understanding and applying probabilities in the field of genetics and how they can aid in
predicting the likelihood of trait inheritance and genotypic outcomes.
3. Genetic Inheritance of a Recessive Trait
In the study of genetics, we often encounter situations where we need to model the probability of specific genetic outcomes. Let's
consider a simple genetic trait, where the presence of a recessive allele (r). An individual can inherit the allele from either parent.
The probability of inheriting a recessive allele from each parent is represented by , where is known to be 0.25.
Part A. Define a random variable that represents the maternally inherited allele. What are the possible outcomes of this random
variable, and what are their probabilities? This random variable is distributed according to what distribution?
Part B. Explain how you can model the inheritance of a recessive allele (r) from both parents using a Binomial distribution. What
does the r.v. represent in this case? What are the parameters of the Binomial distribution in this context, and what do they
represent?
Part C. Calculate the probability that an individual inherits two recessive alleles (rr), one from each from both parents for this trait.
4. Moment Analysis in Measuring Tree Height in a Forest
In forest ecology, understanding the height distribution of trees is crucial for assessing forest structure and biomass. Ecologists
often use probability distributions to describe the distribution of tree heights. Let's consider a forest where tree heights follow a
continuous probability distribution, f(h), where h is the height of a tree in meters.
The probability density function for tree heights in this forest is given by:
where is a constant that ensures the total probability integrates to 1.
Part A: What is the value of ?
Part B: Calculate the mean tree height.
Part C: Calculate the variance in tree height distribution.
Part D. Calculate the third moment (skewness) of the tree height distribution. Discuss what the answers to A, B, and C tell you
about the tree height distribution.
Part E: Plot the distribution of tree heights.
Part F: Suppose you want to compare the tree height distribution of this forest to another forest. How could you use moments
(mean, variance, skewness) to assess and compare the ecological characteristics of these two forests based on tree height data?
5. Random number generation
Part A: Modify the Python code to generate a histogram of 500 exponentially distributed random numbers where . Show
the resulting plot.
Part B: Use the built in function np.random.exponential to generate 500 random numbers from an exponential distribution where
. Show the resulting plot.
Part C: Plot the distributions from part A and part B on the same plot and use the result to compare the two distributions.
6. Challenge Question: Modelling Disease Incubation Periods
Understanding the incubation period of a disease is crucial in epidemiology, as it helps assess disease transmission dynamics and
develop effective control measures. Let's consider a new infectious disease with a known incubation period, during which
individuals are infected, asymptomatic, but still infectious. The incubation period, denoted as follows a continuous probability
distribution with the following probability density function (PDF):
p p
f(h) = khe for 0 ≤
−0.2h h < ∞
k
k
λ = 1
λ = 2
1
T
f
f(t) = 0.2e for t ≥
−0.2t 0
Part A. Calculate the cumulative density function (CDF) of the incubation period, F(t). What does the CDF represent in the context
of this infectious disease, and how can it be useful for epidemiologists?
Part B. Calculate the probability that an individual exposed to the disease will develop symptoms within the first 5 days (0 ≤ t ≤
5). Use the CDF to find this probability.
Part C. Calculate the mean and skew of the incubation period using the PDF. What do these movements imply about disease
transmission and control strategies?
Part D. Discuss the epidemiological relevance of using probability density and cumulative density functions to model disease
incubation periods. How can understanding these functions help in predicting disease spread, estimating outbreak sizes, and
planning public health interventions?