# Darwin-Lamarck

Here are some illustrations to Chapter 11 in Cliff's notes which deals with the Luria-Delbrück Experiment an application of the binomial distribution. Its a nice application of statistics and illustrates how important the Binomial distribution is.**Max Delbrück**and

**Salvador Luria**got the 1969 Nobel Prize in Physiology and Medicine for a 1943 experiment published in a paper

*"Mutations of Bacteria from Virus Sensitivity to Virus Resistance, Genetics 28 (1943), 491-511".*Their experiment was in favor of Darwin's view.

**phage**, a virus attacking bacteria.

Jean-Baptiste Lamarck, 1744-1829Adaptive force: Selection force new traits to arise. |
Here is Lamarck's model. If the survival probability of a virus is p,
then the probability that k bacteria survive is
P[ X = n ] = (N!/(n!(N-n!)) pThis random variable X has the expectation E[X] = N p and the variance Var[X] = N p(1-p) so that Var[X]/E[X] = 1-pwhich is N independent and close to 1. |

Charles Darwin 1809-1882,Natural selection: Selection due to environmental stresses |
Darwin's proposal assumes there is a small probability q that one of
the colonies has a mutation rendering it immune to the virus. Let Y be
the random variable counting the number of colonies with this mutation.
P[ Y = k ] = (K!/(k!(K-k!)) qThe random variable Y has the expectation E[Y] = K q and variance Var[Y] = K q (1-q). But now, if k colonies have found a "cure" to the virus attack, we expect P[X = n] = sumbacteria to survive. The random variable X has now expectation qN and variance q(1-q)N ^{2}, so that
Var[X]/E[X] = (1-q) Nwhich is N dependent and grows exponentially in time. |