• Probability is the expression of belief in some future outcome

• A random variable can take on different values with different probabilities

• The sample space of a random variable is the universe of all possible values

• The sample space can be represented by a

  • probability distribution (for discrete variables)
  • probability density function (PDF - for continuous variables)
  • algebra and calculus are used for each respectively
  • the probabilities of an entire sample space always sum to 1.0
• There are many families or forms of distributions or PDFs
  • depends on the nature of the dynamical system they represent
  • the exact instantiation of the form depends on their parameter values
  • we are often interested in statistics in estimating parameters

• Describes the expected outcome of a single event with probability p

• Example of flipping of a fair coin once

\[Pr(X=\text{Head}) = \frac{1}{2} = 0.5 = p \]

\[Pr(X=\text{Tails}) = \frac{1}{2} = 0.5 = 1 - p \]

• If the coin isn’t fair then \(p \neq 0.5\)
• However, the probabilities still sum to 1

\[ p + (1-p) = 1 \]

• Same is true for other binary possibilities
  • success or failure
  • yes or no answers
  • choosing an allele from a population based upon allele frequences

• A binomial distribution results from the combination of several independent Bernoulli events

• Example - pretend that you flip 20 fair coins and record the number of heads
• Now repeat that process and record the number of heads for each
• We expect that most of the time we will get approximately 10 heads
• Sometimes we get many fewer heads or many more heads

• The distribution of probabilities for each combination of outcomes is

\[\large f(k) = {n \choose k} p^{k} (1-p)^{n-k}\]

• n is the total number of trials
• k is the number of successes
• p is the probability of success
• q is the probability of not success
• For binomial as with the Bernoulli p = 1-q

• Pretend that you want to create your own binomial distribution
  • you could flip 20 fair coins: 20 independent Bernoulli “trials”
  • you could then replicate this 100 times: 100 “observations”
  • for each one of your replicates you record the number of heads
  • probability of heads (“success”) for any flip of a fair coin is 0.5
• Draw a diagram of what you think the distribution would look like
  • what will the x-axis represent, and what are its values?
  • what will the y-axis represent, and what are its values?
  • where will the ‘center of mass’ be?
  • what parameter determines the value of the center of mass?
  • what parameter determines the spread of the distribution?

• Using R, simulate these experimental data
  • using the \(rbinom()\) function
  • look at the arguments to see how they map on to the trials, probability of each trial, and number of replicates
  • use the \(hist()\) function to plot the simulated data.
  • what do the y- and x-axes represent?
  • what is the most common outcome, in terms of the number of “successes” per trial? Does this make sense?
• Now just change you script to do the following
  • perform 200 trials for each of the 100 replicates
  • perform 2000 trials for each of the 100 replicate
  • how do the distributions change when you go from 20 to 200 to 2000 trials?

• Note that the binomial function incorporates both the ‘and’ and ‘or’ rules of probability
• This part is the probability of each outcome (multiplication)

\[\large p^{k} (1-p)^{n-k}\]

This part (called the binomial coefficient) is the number of different ways each combination of outcomes can be achieved (summation)

\[\large {n \choose k}\] Together they equal the probability of a specified number of successes

\[\large f(k) = {n \choose k} p^{k} (1-p)^{n-k}\]

• Now perform your experiment using R
• Use the rbinom function to replicate what you sketched out for coin flipping
• Be sure to check out the other functions in the binom family
• Make sure you know how you are mapping the parameeters to values
• Take the output and create a histogram
• Does it look similar to what you expected?
• Now change the values around so that
  • You have more or fewer coin flips per trial
  • You replicate the process with more or fewer trials
  • You change the coin from being fair to being slightly biased
  • You change the coin from being slightly to heavily biased

• Another common situation in biology is when each trial is discrete but number of observations of each outcome is observed

• Some examples are
  • counts of snails in several plots of land
  • observations of the firing of a neuron in a unit of time
  • count of genes in a genome binned to units of 500 amino acids in length
• Just like before you have ‘successes’, but
  • now you count them for each replicate
  • the replicates now are units of area or time
  • the values can now range from 0 to an arbitrarily large number

• For example, you can examine 100 plots of land
  • count the number of snails in each plot
  • what is the probability of observing a plot with ‘r’ snails is
• Pr(Y=r) is the probability that the number of occurrences of an event y equals a count r in the total number of trials

\[Pr(Y=r) = \frac{e^{-\mu}\mu^r}{r!}\]

• Note that this is a single parameter function because \(\mu = \sigma^2\), and the two together are often just represented by \(\lambda\)

\[Pr(y=r) = \frac{e^{-\lambda}\lambda^r}{r!}\]

• This means that for a variable that is truly Poisson distributed:
  • the mean and variance should be equal to one another, a hypothesis that you can test
  • variables that are approximately Poisson distributed but have a larger variance than mean are often called ‘overdispersed’

where \[\large \pi \approx 3.14159\]

\[\large \epsilon \approx 2.71828\]

To write that a variable (v) is distributed as a normal distribution with mean \(\mu\) and variance \(\sigma^2\), we write the following:

\[\large v \sim \mathcal{N} (\mu,\sigma^2)\]

Estimate of the mean from a single sample

\[\Large \bar{x} = \frac{1}{n}\sum_{i=1}^{n}{x_i} \]

Estimate of the variance from a single sample

\[\Large s^2 = \frac{1}{n-1}\sum_{i=1}^{n}{(x_i - \bar{x})^2} \]

The standard deviation is the square root of the variance

\[\Large s = \sqrt{s^2} \]

• Often we want to make variables more comparable to one another
• For example, consider measuring the leg length of mice and of elephants
• Which animal has longer legs in absolute terms?
• Which has longer legs on average proportional to their body size? Which has more variation proportional to their body size?
• A qood way to answer this last question is to use ‘z-scores’, which are standardized to a mean of 0 and a s.d. of 1
• We can modify any normal distribution to have a mean of 0 and a standard deviation of 1
• Another term for this is the standard normal distribution

\[\huge z_i = \frac{(x_i - \bar{x})}{s}\]

Simulate a population of 10,000 individual values for a variable x:

x <- rnorm(10000, mean=50.5, sd=5.5) 

Take 1000 random samples of size 20, take the mean of each sample, and plot the distribution of these 1000 sample means.

x_sample_means <- NULL
for(i in 1:1000){
x_samp <- sample(x, 20, replace=FALSE)
x_sample_means[i] <- mean(x_samp)
}

For one of your samples, use the equation from the previous slide to transform the values to z-scores.

Plot the distribution of the z-scores, and calculate the mean and the standard deviation.

• Estimation is the process of inferring a population parameter from sample data
• The value of one sample estimate is almost never the same as the population parameter because of random sampling error
• Imagine taking multiple samples, each will be different from the true value
• Most will be close, but some will be far away
• The expected value of a very large number of sample estimates is the value of the parameter being estimated
• Sampling distribution of an estimate
  • all values we might have obtained from our sample
  • probabilities of occurrence
• Standard error of an estimate
  • standard deviation of a sampling distribution
  • measures the precision of the parameter estimate
  • NO ESTIMATE IS USEFUL WITHOUT IT!

x <- c(0.9, 1.2, 1.2, 1.3, 1.4, 1.4, 1.6, 1.6, 2.0, 2.0)

• Use R to make 1000 “pseudo-samples” of size 10 (with replacement), using a for loop as before.
• Name the pseudo-sample object “xboot”, and name the means of the xboot samples “z”.
• Plot the histogram of the resampled means, and calculate the standard deviation of the sample means (the bootstrap SEM) using the sd() function.
• How does it compare with the ordinary standard error of the mean calculated from the original, real sample?

sd(x)/sqrt(10)

• Now take one of the genes from the GacuRNAseq_Subset.csv data and obtain a bootstrapped estimate of the mean expression level.