Probability distribution and Sampling

Probability

• We encounter probability every day in terms of chance:
• What is the chance of rain?
• What is the chance of winning lottery?
• What is the chance of getting a royal flush?

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• Probability theory was born to mathematically define and study the probability

Definitions and Notation

• Sample Space Ω : The set of all possible outcomes of a random experiment.
• Rolling a die once Ω = {1,2,3,4,5,6}

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• Event: Specific outcome or a collection of outcomes from a random experiment. We use capital letters A, B, C, …, to denote events.
• Rolling an even number A = {2,4,6}
• Rolling an odd number B = {1,3,5}

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• Probability of event A occurring Pr(A) = 3/6 = 0.5

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• Probability of A or B Pr(A ∪ B) = 6 / 6 = 1

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• Probability of A and B Pr(A ∩ B) = Pr(A, B) = 0 / 6 = 0

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Independence

• Two events are independent if the outcome of one does not affect the other.

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• Conditional Probability: Pr(A | B) = Probability of event A given that event B has occurred

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• Independence: Pr(A | B) = Pr(A)
• A = Second roll is even B = First roll is odd
• Pr(A | B) = Pr(Second roll is even given that First roll is odd) = Pr(Second roll is even ) =1 / 2 because first row does not affect the second roll

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• If two events A and B are independent, Pr(A, B) = Pr(A | B) Pr(B) = Pr(A) Pr(B).

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Random Variables

Discrete Random Variable

Continuous Random Variable

Human Genome

Single Nucleotide Polymorphism

Central Dogma: DNA -> mRNA -> Protein

Alleles Recode (A as reference)

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AG Aa

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AA AA

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GG aa

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Sequencing Reads

Examples of Random Variables in Biology

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• The number of reference allele of a genetic variant of an individual
• X = 0 if individual carries homozygous for the reference allele
• X = 1 if individual carries heterozygous for the reference allele
• X = 2 if individual carries homozygous for the alternative allele

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• The number of sequencing reads mapped to a gene
• X = 1000 if there are 1000 reads mapped to a gene

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• The status of having diabetes of an individual
• X = 1 if individual has diabetes, X = 0 if individual is healthy

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• Always define your random variable before using it!

Frequentist View of Probability

• More broadly, you can think of nature performing experiments to generate random variables, and we observe the generated random variables.

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• Nature repeats experiments at infinitely times, independently, under the same condition, the proportion of times the random variable takes a value is probability.

Monte Carlo Simulation

• Use computer to perform experiments to generate random variables

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• We can repeat the experiments for a very large number of times, approach to infinity

Probability Distribution

Bernoulli distribution

Binomial distribution

Hardy-Weinberg Principle in Population Genetics

Cumulative Distribution Function (CDF)

Cumulative Distribution Function (CDF)

Other family of discrete distribution

• Multinomial distribution: Generalization of binomial to multiple categories.

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• Hypergeometry distribution: Number of success in a fixed number of draws without replacement in a finite population.
• Ex: 3 lottery-winning cards in a pot of 52 cards. If I draw 15 cards without replacement, what is the probability of getting at least one lotter—winning card?

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• Poisson distribution/ Negative binomial distribution (Gene expression data RNA-seq/single cell RNA-seq)

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Poisson distribution

The number of times an event occurs in a fixed interval of time or space with rate λ.

Connection to Binomial distribution: n is large and p is small, λ = p/n.

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Mean = Variance (defined in next week’s lecture)

Luria-Delbruck Experiment (1969 Nobel Prize in Medicine)

Do mutations arise randomly before selection (Darwinian), or are they induced by the environment (Lamarckian)?

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Technical replicates: Sequence same samples in different lanes and runs

Negative Binomial

Slides from Terry Speed

Biological replicates: Repeat experiments

Variance = Mean + a * Mean^2 (Overdispersion)

Continuous random variable

Probability density function

Intuition of integration

Height: f(t)

Area of rectangle: f(t) dt

Normal distribution

• Due to the Central Limit Theorem (covered later), the normal distribution (“bell curve”) is often observed in real data.

Normal distribution

Other family of continuous distribution

• Uniform distribution

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• t distribution (Week 6)

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• Chi square distribution (Week 6)

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• F distribution (Week 7)

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