WITH PYTHON PLEASE

^{Question ( Simulations for Normal Distribution):}

^{We said that normal distribution typically results in when we have the addition of many occurrence of another random phenomena. In this simulation, we will add many uniform distribution random variables with the same mean and the variance, and will observe that we will get normal distribution. For this perform the followings:}

^{(a) (5 Pts.) Generate a uniform random variable with mean c1 and standard deviation c4.}

^{(b) (5 Pts.) Generate 10 000 of the samples of the uniform distribution given in Part (a). First, add these random samples and then divide the total sum by the number of samples, or 10 000. This will give you one outcome for a normal distributed random variable. In order to collect many samples, repeat this part by 1 000 000 times.}

^{(c) (5 Pts.) Determine the histogram, PMF, and the PDF for the results obtained in Part (b). Clearly define your bin width for each of these plots.}

^{(d) (5 Pts.) Comment on what you observe for Part (c) in terms of the PDF shape, the mean, and the variance. Draw the theoretical normal distribution PDF (i.e., from the formula) on top of your PDF and observe that they match.}

^{(e) (10 Pts.) If we represent this new normally distributed random variable as X, simulation- wise determine P(c1X + c2 < c4). Then, theoretically determine this probability and}

^{observe that they match. If they do not match, comment on why.}

^{Note: C1=5, C2=4, C4=9}