Physics 380, 2011: Lecture 3
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Back to Physics 380, 2011: Information Processing in Biology.
We are continuing our review of some basic concepts of probability theory, such as probability distributions, conditionals, marginals, expectations, etc. We will discuss the central limit theorem and will derive some properties of random walks. Finally, we will study some specific useful probability distributions. In the course of this whole lecture block, we should be thinking about E. coli chemotaxis in the background -- all of these concepts will be applicable.
A very good introduction to probability theory can be found in Introduction to Probability by CM Grinstead and JL Snell.
Warmup question
- The famous Luria-Delbruck (1943) experiment has shown that mutations appear at random in bacteria, rather than directly in response to an environmental pressure. In the experiment, the grew a small number of bacteria in culture tubes, and then plated samples from these tubes onto agar and provided stress with application of a phage. If bacteria have a low probability of directly responding to a phage, what should a distribution of the number of colonies that survive the phage application be?
Main lecture
- Multivariate distributions
- Conditional and joint probabilities, Bayes theorem: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A,B)=P(A|B)P(B)=P(B|A)P(A)}
- independence: two variables are independent if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A,B)=P(A)P(B)} , or, equivalently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A|B)=P(A)} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(B|A)=P(B)} .
- Expected values: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(f(x))=\int_{-\infty}^{+\infty}f(x)P(x)dx}
. In particular, a few of the expectation values are very common: the mean, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x\rangle=\mu=\int_{-\infty}^{+\infty}xP(x)dx}
, and the variance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x^2\rangle-\langle x\rangle^2=\sigma^2=\int_{-\infty}^{+\infty}x^2P(x)dx - \mu^2}
.
- addition of independent variables: in general, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(f(x)+g(x))=E(f(x))+E(g(x))} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(f(x)g(y))=E(f(x))E(g(y))} , provided Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} are independent, that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x,y)=P(x)P(y)} .
- Moments, central moments, and cumulants -- What are the moments of the E. coli trajectory?
- Cumulants: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1=\mu} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=\sigma^2} , and higher order cumulants measure the difference of the distribution from a Gaussian (all higher cumulants for a Gaussian are zero)
- Moment and cumulant generating functional
- Moment generating function (MGF): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_x(t)=E(e^{tx})} . The utility of MGF comes from the following result: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_n=\left.\frac{d^n M_x(t)}{dt^n}\right|_{t=0}} . That is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(t)=1+ \mu_1t+\frac{\mu_2}{2}t^2+\frac{\mu_3}{3!}t^3+\cdots} .
- Properties of MGF:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{x+a}(t)=e^{at}M_x(t)} . From this we can show that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(y|x)=P_1(y-x)} , that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_3(y)=\int P_1(y-x)P_2(x)} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_3(t)=M_1(t)M_2(t)} . Alternatively, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=x+y} , then the MGF of z is a product of the MGF's of x and y.
- Cumulant generating function (CGF): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_x(t)=\log M_x(t)} . Then the cumulants are: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_n=\left.\frac{d^n C_x(t)}{dt^n}\right|_{t=0}} . That is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(t)=c_1t+\frac{c_2}{2}t^2+\frac{c_3}{3!}t^3+\cdots} . CGF of independent variables add.
- Using the addition of CGFs, we show that when random variables add, their means and variances add.
- Why do measurements of a quantity many times improve the measurement?" Frequencies and probabilities: Law of large numbers. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\frac{1}{n}\sum x_i} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_S=\mu_x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2_S=\sigma^2_x/n} . This follows from the addition of means and variances. So, we can calculate the mean and the variance of the E. coli motion.
