8 [MATLAB exercise] The MA'I'LAB function 'randn' is designed to generate samples of a random variable, say X , which is governed by a unit-Gaussian p.d.f. (i.e., a Gaussian p.d.f. with px = 0 and a} = 1). In this exercise, we will investigate the ability of the routine to generate truly Gaussian samples. (a) Let X be a unit-Gaussian random variable. Using pencil and paper, compute P(0 S X S 0.5) and P(0.9 S X g 1). (b) Use the MATLAB function randn to generate 1000 samples of a unit-Gaussian random variable Let's call the vector X. (c) Use the MATLAB function 'histogram' to create a histogram of the values in X with a bin width of 0.05. ((1) Plot the histogram and comment on the approximation to a unit-Gaussian p.d.f. (9) Use the histogram data to estimate the two probabilities computed in part (a). For example, No. samples in the bin covering the range 0 S X < 0.05 < < . P(0 X 0 5) *5 Total no. values : 1000 Comment on the result. (f) Repeat the steps (b)(e) for a 10000 point sample of unitGaussian data. (g) Does the Gaussian approximation improve as the number of samples increases? Consider the bivariate joint PDF of X and Y given by (c is some constant) fxy (u, v) = otherwise 1. Find the value of c. 2. Find p(X Sys 3. Find fy(v) the marginal distribution of Y. 4. Compute p(Y < ;) 5. Compute E(Y) and V(Y) 6. Find fxy(u; v) the conditional distribution of X given that Y = v 7. Find E(X | Y= 8. Find p(X > |Y = ) 9. Determine whether X and Y are independent.Question 4 E - EEbond + [Ears + EEtors + EEww + > Electrostatic bonds angles fortions anit pairs aim pairs A general equation for a molecular mechanics force field is shown above: a) Explain why molecular mechanics force fields are described as being 'empirical". b) Explain how Ebond is calculated. C) Explain how Eelectromeans is calculated. d) Explain how E. is calculated. e) The calculation Eswaranats is the most time consuming part of molecular mechanics calculations, Explain why this is so. Explain how the amount of computation necessary to calculate Execraware can be reduced. What problems can these approximations cause? D) Describe two approximations that are made to speed up molecular mechanics calculations.Question 3 [10 marks] The molecular mechanics computational technique makes use of parametristd force fields to perform calculations. Briefly describe the main features of the molecular mechanics method, explaining the term "parametriced force fields". Use an illustration in your answer [3 marks] b) Most modern molecular mechanics methods employ a four-component model to yield the total energy - what are the four components? [4 marks] (c) Briefly describe how molecular mechanics can be used as a tool in drug discovery. [3 marks]