Lecture 4: Probability Distributions PDF

LECTURE 4 September 17, 2024 Mean and Variance Sample mean 1 𝑚=µ= 𝑥1 + 𝑥2 + … + 𝑥𝑁 𝑁 Sample Variance 1 𝑆2 = [ 𝑥 −𝑚 2 + … + 𝑥𝑁 − 𝑚 2 ]...

LECTURE 4 September 17, 2024 Mean and Variance Sample mean 1 𝑚=µ= 𝑥1 + 𝑥2 + … + 𝑥𝑁 𝑁 Sample Variance 1 𝑆2 = [ 𝑥 −𝑚 2 + … + 𝑥𝑁 − 𝑚 2 ] 𝑁−1 1  Expected value 𝑚 = 𝐸 𝑥 = 𝑝1 𝑥1 + 𝑝2 𝑥2 + … + 𝑝𝑁 𝑥𝑁 ) Variance 1 𝛔2 = [ 𝑥1 − 𝑚 2 + … + 𝑥𝑁 − 𝑚 2 ] 𝑁−1 Standard deviation Sample Space Set of all possible outcomes of an experiment – Coin = {H,T} – Die = {1,2,3,4,5,6} – What about two coins? – What about three coins? – What about the sum of rolling two dice? Probability The likelihood of an event occurring Tossing a coin (1/2) Rolling a dice (1/6) What is the probability of the name Jon in the being drawn from a list of names in the room? What is the probability of rolling a total of 2 on two dice? What is the probability of rolling a total of 1 on two dice? What is the probability of rolling a total of 11 on two dice? Probability Independence – 𝑃 𝐸𝐹 = 𝑃 𝐸 𝑃 𝐹 Conditional Probabilities – 𝑃(𝐸|𝐹) = 𝑃(𝐸𝐹)/𝑃(𝐹) Probability Distributions Distribution Type Examples Binomial Tossing a coin n times Poisson Rare events Exponential Forgetting the past Gaussian = Normal Averages of many tries Log-normal Logarithm has normal distribution Chi-squared Distance squared in n dimensions Multivariable Gaussian Probabilities for a vector Normal Distribution Many biological and physiological measurements, such as height, weight, blood pressure, etc, have a normal distribution Central limit theorem – As N goes to infinity, the sample mean will be approximately normally distributed. Skewness – Positive skewness indicates a longer right tail – Negative skewness indicates a longer left tail Poisson Distribution Used to describe the distribution of rare events in a large population. – For example: How often a cell within a large population of cells will acquire a mutation Exponential Distribution Used to model random events occurring over time at a constant rate – For example, the decay of a protein or the rate of mutations on a DNA strand Binomial distribution A discrete probability distribution that can be used to calculate the probability of a certain number of successes in a given number of trials How do we compare samples? Odds ratio A statistic that quantifies the strength of the association between two events Odds of exposure in cases – The number of cases with exposure compared to the number of cases without exposure Odds of exposure in controls – The number of controls with exposure compared to the number of controls without exposure t-test An inferential statistic used to determine if there is a significant difference between the means of two groups Paired (Dependent) T-test Equal Variance (Independent) T-test Unequal Variance (Independent) T-test Multiple hypothesis testing What happens when you are testing multiple things at the same time? – Gene expression – Mutational burden of genes – GO term analysis Multiple testing correction Bonferroni – ⍺/m Benjamini-Hochberg procedure – Sort the p-values in ascending order – (⍺/m) * rank