You must know that …

• An experiment is a random process
  • We don’t know the result until we do the experiment
• An experiment produces a single outcome
  • Cells survival, concentration of DNA, temperature
  • Outcome may be logic, factor, numeric or character

You must know that …

• Replicating the experiment produces a sample

  • A sample is a vector

• Experiments produce samples

• We care about populations

You must know that …

Populations are BIG.

• Like “all people in the planet” or “all experiments in all parallel universes”

• In the class we use known populations

• In real life populations are unknown (partially)

You must understand …

• Experiments give samples, but we care about populations

• What happens in the sample depends on the population

You must understand …

• By looking at the samples we can learn about the population

• Populations are big

  • We assume they have infinite size to make calculation easier

You must know how to …

• Simulate experiments using the sample() function

• Prepare the outcomes vector

• Use sample(outcomes, size=n) to get n random elements from outcomes

You must know that …

• Most times (but not always) we use replace=TRUE

• This allows size bigger than length(outcomes)

• More important: probabilities do not change

You must know that …

• When population is small, sampling will change the population
  • We do not do this in this course
• When population is big, sampling has a very small effect
  • The difference between 1E8 and 1E8 -1 is not important

You must know that …

• When populations have infinite size, taking a sample has no effect

• When we replace the sample, there is no effect on the population

• Sampling with replacement is the same as having an infinite population

You must know that …

• When we use sample, each outcome can have a different probability
• The probability distribution is a vector with the same length as outcomes
• We can use the option prob= to change the probability distribution
• If we do not use prob=, then all outcomes have the same probability

You must know how to …

• Decompose a complex random process in smaller parts
• Simulate a complex random system and find the empirical frequencies
• Draw the results in a bar plot
• Use the option prob=p to change how sample() works

You must know that …

• These simulations are called “Monte-Carlo Methods”

• This allow us to explore cases that have too many combinations

You must know that …

• We cannot see all possible genes of length 1000bp

• There are 4¹⁰⁰⁰ = 2²⁰⁰⁰ = 2^10x200 ≈ 10^3x200 = 10⁶⁰⁰ combinations

• The age of the universe is ≈ 4.32x10¹⁷ seconds

You must understand …

• Testing all possible cases is impossible

• Random sampling allows us to get an idea of all possible cases

• More simulations give better approximations, but take longer time

• This is one of the most common uses of computers in Science

Events: You must know that …

• An event is any logical question about the experiment outcome
  • such as “two people have the same birthday”
• In R we can use functions, taking an outcome and returning TRUE or FALSE

Events: You must know that …

Outcomes and events are different

• An event can be true for several different outcomes

• An experiment produces only one outcome, and several events

Events: You must know how to …

Write a function to represent an event

• Takes a sample (one or more outcomes) as input
• Returns TRUE or FALSE depending on the event rule
• For example:
  • Two people having the same birthday
  • A student passes this course

You must know that …

• A random variable is any numeric value that depends on the experiment outcome
  • such as “the number of people with epilepsy in our course”
• In R we can use a numeric outcomes vector

You must know that …

• In this case there is a
  • population average,
  • population variance, and
  • population standard deviation
• In general we do not know the population average, and we want to know it

You must know that …

The population standard deviation measures the population width

Chebyshev theorem says \[ℙ(|x_i-\bar{\mathbf x}|≥ k⋅\text{sd}(\mathbf x))≤ 1/k^2\] It can also be written as \[ℙ(|x_i-\bar{\mathbf x}|≤ k⋅\text{sd}(\mathbf x))≥ 1-1/k^2\]

Chebyshev: You must know how to …

Find the population width for different values of \(k\)

At least 75% of the population is near the average, by no more than 2 times the standard deviation \[ℙ(|x_i-\bar{\mathbf{x}}|≤ 2⋅\text{sd}(\mathbf x))≥ 1-1/2^2\] \[ℙ(\bar{\mathbf{x}} -2⋅\text{sd}(\mathbf x)≤ x_i ≤ \bar{\mathbf{x}} +2⋅\text{sd}(\mathbf x)) ≥ 0.75\]

Chebyshev: You must know how to …

At least 88.9% of the population is near the average, by less than 3 times the standard deviation \[ℙ(\bar{\mathbf{x}} -3⋅\text{sd}(\mathbf x)≤ x_i ≤ \bar{\mathbf{x}} +3⋅\text{sd}(\mathbf x)) ≥ 0.889\]

You must understand …

Everything we measure will be in an interval

The interval depends on the population standard deviation and the confidence level

You must understand …

Chebyshev theorem is always true, but in some cases is pessimistic

In some cases we can have better confidence levels

You must know that …

• You can take the average of a sample to estimate the population average

• The sample average is a random variable. Changes on every experiment

You must know that …

• When the sample is bigger, the sample average will be closer to the population average
  • This is called Law of Large Numbers
• Moreover, the sample average of a big sample will follow a Normal distribution
  • This is called Central Limit Theorem

You must understand that …

• The Chebyshev rule is always valid, but pessimist
  • confidence intervals are big
• If the probability distribution is Normal, we can have better confidence intervals

You must understand that …

• Not all probabilities are Normal
• When the random process is a sum of many parts, then we may have a Normal distribution
  • That happens with experimental measurements
  • Also happens in many biological processes

The problem with confidence

You may have noticed that we never get 100% confidence

That is a fact of life. We have to accept

To have very high confidence, we need wide intervals

But wide intervals are less useful