## Lecture 2.2

**Title:** Tutorial — Basics of Probability Distributions and Statistics for Single-Cell Data

**Lecturer: **Dr. Huy Vo

**Lecturer Website: https://www.engr.colostate.edu/~munsky/**

**Lecturer Email**: huy.vo@colostate.edu

**Learning Objectives:**

Learn how to do some basic statistical analyses in Python

**Dr. Huy Vo** is a postdoctoral researcher in the Munsky Group at Colorado State University. He earned a Ph.D. degree in Mathematics at the University of Alabama in 2017. His current research focuses on developing new computational tools to design informative single-cell experiments that account for both intrinsic noise and measurement uncertainty. Other interests: parameter estimation, uncertainty quantification, model reduction for stochastic gene expression models, software development.

**Title:** Basics of Probability Distributions and Statistics for Single-Cell Data

**Abstract:** abc abc abc

**Suggested Reading or Key Publications:**

- Explore basic univariate probability distributions: Colab Link
- Visualization and summary statistics for multivariate and time-series data. Colab Link
- Linear Regression: estimating free diffusion coefficient from mean squared displacements. Colab Link

**Links to Relevant Software: **

- e
- f

- A Gene is ON at some time 𝑡=0. It can turn OFF at a stochastic rate of 5/min*. It can also create one mRNA at a time at a stochastic rate of 20/min. What is the distribution of mRNA created before the gene turns OFF?
- A Gene is ON at some time 𝑡=0. It later turns OFF at exactly 𝑡=1/5 min. It can also create one mRNA at a time at a stochastic rate of 20/min. What is the distribution of mRNA created before the gene turns OFF?
- Why are the two random variables above different? Which is more variable?
- Consider 2 genes that are both ON and both can turn OFF with a stochastic rate of 5/min.
- What is the distribution of time until the first of these genes turn OFF?
- What is the distribution of time until BOTH of these genes turn OFF?

- A Gene is ON at some time t=0. It can turn OFF at a stochastic rate of 5/min. What is the probability that it is still ON at a time t = 1 min?
- What is the Fano Factor of a Poisson random variable? Of an exponential random variable? How do these depend on the mean of the random variables?
- What is the Coefficient of Variation (std/mean) of a Poisson random variable? Of an exponential random variable? How do these depend on the mean of the random variable?
- Consider two independent normal distributed random variables both with mean of 2 and a standard deviation of 1. What is the distribution of the sum of pairs of these two random variables? What is its mean? What is its standard deviation?
- Consider two identical (non-independent) normal distributed random variables both with mean of 2 and a standard deviation of 1. What is the distribution of the sum of pairs of these two random variables? What is its mean? What is its standard deviation?
- When will the Central Limit Theorem fail to work?