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01:960:285 Lecture Notes - Lecture 12: Joint Probability Distribution, Conditional Probability, Sample Space

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School
Rutgers University
Department
Statistics and Biostatistic
Course
01:960:285
Professor
Shyam Moondra

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Document Summary

Recall that the probability of an event occurring given that another event has already occurred is called a conditional probability. The probability that event b occurs, given that event a has already occurred is. This formula comes from the general multiplication principle and a little bit of algebra. Since we are given that event a has occurred, we have a reduced sample space. Instead of the entire sample space s, we now have a sample space of a since we know. So the old rule about being the number in the event divided by the number in the sample space still applies. It is the number in a and b (must be in a since a has occurred) divided by the number in a. If you then divided numerator and denominator of the right hand side by the number in the sample space s, then you have the probability of a and b divided by the probability of a.

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Related Questions

You are given a fair die, i.e. the outcomes of any throw are contained in the sample space   Let the event A be that an even number is thrown and the event B that a multiple of three is thrown. Now consider the two random variables X and Y. X takes on a value of 1 if the event A has occurred and the value 0 if A has not occurred. The random variable Y takes on the value of 1 if the event B has occurred and the value zero if B has not occurred.

(a) Derive the joint pdf of X and Y. Provide this in the form of a table

                                        Y=0                                Y=1

X=0    
X=1    

b) What is the marginal probability distribution of Y?
(c) What is the conditional probability distribution of X, given that Y = 1?
(d) What is E (X|Y = 1)?
(e) What is Cov (X, Y )?
(f) Are X and Y independent random variables?

Recombinant DNA Technology

The questions all relate to the same topic and somewhat interrelate

1. Plasmids used in vitro to clone foreign DNA fragments are called ___

A. Transgenic.

B. Clones.

C. Vectors.

D. Conjugants.

2. VNTR alleles are hypervariable regions of human DNA that differ from each other in:

A. Location of internal sites recognized by restriction enzymes.

B. Variable number of point mutations.

C. Number of copies of an internally repeated DNA sequence.

D. Variable location on different chromosomes.

3. Probability calculations are used in forensic applications of DNA fingerprinting to determine if:

A. DNA from two different sources has matching alleles.

B. DNA samples were degraded before analysis.

C. A match between alleles of different DNA samples might have occurred by chance.

D. An unknown DNA sample is from a male or a female.

4. Plasmid vectors used in cloning often contain a gene for the N-terminal 146 amino acids of the enzyme β-galactosidase. What is the purpose of including this gene in the vector?

A. It allows selection of E. coli host cells that contain the plasmid.

B. It allows selection of E. coli host cells that contain plasmid in which the insert has been ligated.

C. It cleaves the insert to allow it to be ligated into the vector.

D. It enables the plasmid vector to replicate in E. coli host cells.

5.The first gene therapy clinical trial took place in 1990, on patients with SCID (severe combined immunodeficiency syndrome). Is this true or false?

A. True

B. False

1. Assuming that the average rate of neutral mutation is 10^-9per base pair per gamete, how many generations would it take, onaverage, for 20 base pair substitutions tobe fixed ina gene with200 base pairs? Suppose that the number of base pair differences inthis gene between species A and B is 92, between A and C is 49, andbetween B and C is 91. Assuming that no repeated replacements haveoccurred at any site in any lineage, draw the phylogenetic tree,estimate the number of fixations that have occurred along eachbranch, and estimate the number of generations since each of thetwo speciation events.



2. If a recessive lethal allel has a frequency of 0.050 in newlyformed zygotes in one generation, and the locus is inHardy-Weinberg equilibrium, what will be the allele frequency andthe genotype frequencies at this locus at the beginning of the nextgeneration? (Answer: q=0.048, P^2=0.9071, 2pq=0.0907, q^2=0.0023).Calculate these values for the succeeding generation. If the lethalallele arises by mutation at rate of 10^-6 per gamete, what will beits frequency at equilibrium? (Hint: The last questions is a trickquestion)



Please, help me.