2. Brian is monitoring the number of users playing Vegetable Crossing after the latest round of bug fixes. He finds that the number of active users on a given day can be modeled as a normal distribution. Remember that the probability density function for a normal distribution with mean i and standard deviation o is 1 e-(3-1)/(20%) 27 Brian finds that the mean number of active users on a given day is 800 thousand and that the standard deviation is 100 thousand. (a) Write down a formula for a probability density function pf), with 3 (measured in thou- sands) being the number of active users on a given day, (b) Sketch a graph of p(x). You may use a graphing aid such as Desmos if you like. What features of your graph suggest that this is a realistic model for the number of active users on a given day? What features are unrealistic? (c) Use the known Taylor series for e" centered around x = 0 to write a formula for the Taylor series for p(x) around 2 = 800. Make sure to give the first four nonzero terms and also write the entire series using summation notation. (You may find it useful to know that if Cmx" is the Taylor series of a function f(x) around 2 = 0, then Ch(2-a)" is the Taylor series of f(-a) around 3 = a.) (a) Use your answer to find a cumulative density function P(x) for this distribution. Make sure to write out the first four nonzero terms for P(x), and to write the entire series using summation notation. (e) Use the degree 5 Taylor polynomial for P.) to approximate answer in the context of this problem (1) Find p(2020)(800). no TO Spon plat) dr. Interpret your 2. Brian is monitoring the number of users playing Vegetable Crossing after the latest round of bug fixes. He finds that the number of active users on a given day can be modeled as a normal distribution. Remember that the probability density function for a normal distribution with mean i and standard deviation o is 1 e-(3-1)/(20%) 27 Brian finds that the mean number of active users on a given day is 800 thousand and that the standard deviation is 100 thousand. (a) Write down a formula for a probability density function pf), with 3 (measured in thou- sands) being the number of active users on a given day, (b) Sketch a graph of p(x). You may use a graphing aid such as Desmos if you like. What features of your graph suggest that this is a realistic model for the number of active users on a given day? What features are unrealistic? (c) Use the known Taylor series for e" centered around x = 0 to write a formula for the Taylor series for p(x) around 2 = 800. Make sure to give the first four nonzero terms and also write the entire series using summation notation. (You may find it useful to know that if Cmx" is the Taylor series of a function f(x) around 2 = 0, then Ch(2-a)" is the Taylor series of f(-a) around 3 = a.) (a) Use your answer to find a cumulative density function P(x) for this distribution. Make sure to write out the first four nonzero terms for P(x), and to write the entire series using summation notation. (e) Use the degree 5 Taylor polynomial for P.) to approximate answer in the context of this problem (1) Find p(2020)(800). no TO Spon plat) dr. Interpret your