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4 15 Pts Common Operation Many Computer Simulations Generation Samples Discrete Random Var Q43907408

4. (15 pts.) A common operation in many computer simulations is the generation of samples from a discrete random variable. The problem essentially can be mathematically reduced to the following simple case. Consider a discrete random variable x taking values in the set Sx = {0,1,2,…,n-1 with probability mass function (PMF) px, so that Px() is the probability that the random variable X takes value xe Sx. Let Fx be the cumulative distribution function (CDF) of X, whose domain is the set of real values R = (-0, +00), and is defined as Fx (x) = P(XS) = if : <0, .px(), if : € 0,9-1], if >n-1. A simple general approach to generate random samples from X according to its PMF PX is to draw a number r uniformly at random from the interval (0,1), and then use the CDF FX to select the value x as the random sample of X if Fx (x – 1) Sr <Fx(a). The following is pseudocode for two implementations of this approach. Both algorithms take as input an n-element array P such that each element Pi = px) for all i=0,1,….n-1, and outputs an integer value a between 0 and n-1 (inclusive). Assume that the function RAND ouputs a random number uniformly distributed between 0 and 1 in constant time using a constant amount of space, Algorithm 1 RAND DISCRETEI(P) 1: F new n-element array 2: FO + PO 3: for 1 to n – 1 do 4: FG] – PO 5 for it to j do 6: Fly Fli] + P[i] 7: RANDO 8: 20 9: while r > F[2] do 10: 2+1 11: return Algorithm 2 RANDDISCRETE2(P) 1: F new n-element array 2: F0 – PO 3. for 3 + 1 to n – 1 do 4: Flui Flj – 1] +PG] 5: T + RANDO 6: 10 7: while r > F[x] do 8: + 2+1 9: return (a) Give the tightest running-time and space characterization using big-Oh and big-Omega, or big-Theta, of RAND DISCRETEI in terms of n. Briefly justify your answer. Time Space (b) Give the tightest running-time and space characterization using Big-Oh and Big-Omega, or Big-Theta, of RAND DISCRETE2 in terms of n. Briefly justify your answer. Time Space (c) Which algorithm do you recommend to implement, and why? Briefly justify your answer. (d) Write, in pseudocode, a linear-time algorithm RAND DISCRETE3 for this computational problem that takes an array P of n elements as input and outputs a random integer consistent with a discrete random variable with integer values 0,1,…,-1 according to the probability mass function encoded by P and that uses a constant amount of space. Is this algorithm better than the algorithm you recommended to implement in the previous part of this problem? Please explain briefly. Show transcribed image text 4. (15 pts.) A common operation in many computer simulations is the generation of samples from a discrete random variable. The problem essentially can be mathematically reduced to the following simple case. Consider a discrete random variable x taking values in the set Sx = {0,1,2,…,n-1 with probability mass function (PMF) px, so that Px() is the probability that the random variable X takes value xe Sx. Let Fx be the cumulative distribution function (CDF) of X, whose domain is the set of real values R = (-0, +00), and is defined as Fx (x) = P(XS) = if : n-1. A simple general approach to generate random samples from X according to its PMF PX is to draw a number r uniformly at random from the interval (0,1), and then use the CDF FX to select the value x as the random sample of X if Fx (x – 1) Sr F[2] do 10: 2+1 11: return
Algorithm 2 RANDDISCRETE2(P) 1: F new n-element array 2: F0 – PO 3. for 3 + 1 to n – 1 do 4: Flui Flj – 1] +PG] 5: T + RANDO 6: 10 7: while r > F[x] do 8: + 2+1 9: return (a) Give the tightest running-time and space characterization using big-Oh and big-Omega, or big-Theta, of RAND DISCRETEI in terms of n. Briefly justify your answer. Time Space (b) Give the tightest running-time and space characterization using Big-Oh and Big-Omega, or Big-Theta, of RAND DISCRETE2 in terms of n. Briefly justify your answer. Time Space
(c) Which algorithm do you recommend to implement, and why? Briefly justify your answer. (d) Write, in pseudocode, a linear-time algorithm RAND DISCRETE3 for this computational problem that takes an array P of n elements as input and outputs a random integer consistent with a discrete random variable with integer values 0,1,…,-1 according to the probability mass function encoded by P and that uses a constant amount of space. Is this algorithm better than the algorithm you recommended to implement in the previous part of this problem? Please explain briefly.

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Answer to 4. (15 pts.) A common operation in many computer simulations is the generation of samples from a discrete random variabl…