Exercise 2 - Introduction to Probability theory

These exercises cover the sections of LMS_Statistics

Set working directory

# Set working directory
setwd("/Volumes/bioinfomatics$/jurtasun/Courses/CBW2022/LMS_Statistics/course/exercises")
getwd()

## [1] "/Volumes/bioinfomatics$/jurtasun/Courses/CBW2022/LMS_Statistics/course/exercises"

Generating events in R

• rbinom(): generate Binomial distributed random variables with a given mean and std
• rpois(): generate Poisson distributed random variables with a given mean and std
• rnorm(): generate Normal distributed random variables with a given mean and std

Four functionn for simulating data following each distribution

• r: random number generation (rbinom, rpois, rnorm)
• d: for density (dbinom, dpois, dnorm)
• p: evaluate cumulative distribution (pbinom, ppois, pnorm)
• q: obtain quantile function (qbinom, qpois, qnorm)

Exercise 1

• Simulate 10 flips of a coin, and compute the probability of obtaining 5 heads

• Hint: use the rbinom() function

• According to a survey, 72% of Americans prefer dogs to cats. If 8 people are chosen randomly, what is the probability that 6 prefer dogs? And the probability that less than 6?

• Hint: use the dbinom() and pbinom() function

• A weighted coin has 42% chance coming up heads. What is the expected number of heads in 5 tosses? Compute mean and std.

Exercise 2

• Calls to a customer service line at average rate of 6 every 5 minutes. What is the probability of getting exactly 4 calls in 5 minutes? And at least 4?

• Hint: use the dpois() and `ppois()``function

• Compute the probability of reporting 15 or less cancer patients in a given time interval, assuming the historical average is 12

• Compute the probability of reporting 15 or more cancer patients in a given time interval, assuming the historical average is 12

• Hint: use the rpois() function

Exercise 3

• Compute the probability of a value being less than or equal to 2 for a normal distribution of mean 0 and standard deviation 1

• Compute the probability of a value being greater than 2 for a normal distribution of mean 0 and standard deviation 1

• Hint: use the pnorm() function

• Data visualization - generate a gaussian distribution

• Set mean and standard deviation to plot a normal distribution

bonus question 1 (optional)

• Load the gene expression matrix that has been created for this exercise from "data/gene_exp_matrix.RData"

• Use the gene expression to draw a heatmap

• Use the scale() function to perform the Z-score transformation, and use the code above to generate the scaled heatmap

• Hint: read the help page of ?scale, and you might need to use t() function as well

However, the heatmap.2() function has a argument called scale, which does the same thing for you….

bonus question 2 (optional)

• Read in the file "categories_Expression.txt"

• How many genes are in ofInterest and pathway sections?

• Get the quantiles of overall Expression, and for the Glycolysis and TGFb genes

• Find how many genes were selected and in the Glycolysis pathway

• Compute probability of selecting a gene with at least the expression level of "Gene13", assuming normal distributed data

• Perform a t-test to evaluate the difference of the Expression levels between genes in the Glycolysis pathway and genes in the TGFb pathway.