Assignment 2 Due 9 February 2023 indicates problems for students in Chem 608. 1. Consider the function eax with 0x. (a) Turn this function into a probability distribution, Pe(x). (b) Compute the expectation value x. (c) Sketch the distribution and mark the value of eax at x. 2. Find the first and second moments of the uniform distribution, Pu(x)= 1/a. 3. What is the average value of x, given a distribution function q(x)=cx, where x ranges from zero to one, and q(x) is normalized? 4. Histograms and probability distributions. (a) In a jupyter notebook, use one of the libraries you learned about in PS1 to sample the Poisson probability distribution(s) in Barlow and make a histogram. (b) Vary the parameters and confirm his plots. 5. Does the mean match the most probable value for (a) the Gaussian distribution or (b) the Poisson distribution? Justify your answers. 6. You are comparing protein amino acid sequences. You have a 20-letter alphabet ( 20 different amino acids). Each sequence is a string n letters in length. You have one test sequence and s different data base sequences. You may find any one of the 20 different amino acids at any position in the sequence, independent of what you find at any other position. Let p represent the probability that there will be a 'match' at a given position in the two sequences. (a) In terms of s,p, and n, how many of the s sequences will be perfect matches (identical residues at every position)? (b) How many of the s comparisons (of the test sequence against each database sequence) will have exactly one mismatch at any position in the sequences? 7." Confirm the Gaussian distribution is the limit of binomial numerically. 8." Confirm the law of large numbers numerically. Extra credit. Prove the weak law of large numbers, Pr(XX)=0. Assume the variance is finite. Hint; Use Chebyshev's inequality and the variance sum theorem