Exponential Distributions

Exponential distributions

from matplotlib import pyplot as plt

from symbulate import *

W = RV(Exponential(rate = 1 / 30))

w = W.sim(10000)

w

Index	Result
0	19.31839707102859
1	38.907071499553666
2	21.927303602168685
3	15.075911145136045
4	14.569965399393402
5	68.87110665884259
6	22.488723125132488
7	2.264654223306332
8	26.614583774147018
...	...
9999	33.17331598333397

plt.figure();

w.plot()

Exponential(rate = 1 / 30).plot()

plt.show();

w.count_gt(45) / 10000, 1 - Exponential(rate = 1 / 30).cdf(45)

(0.2227, 0.2231301601484298)

Scaling

U = RV(Exponential(rate = 1))

W = U / (1 / 30)

plt.figure(); 

W.sim(10000).plot()

Exponential(rate = 1 / 30).plot()

plt.show();

U = RV(Exponential(rate = 1))

V = U / 2

plt.figure(); 

V.sim(10000).plot()

Exponential(rate = 2).plot()

plt.show();

Memoryless property

plt.figure();

W.sim(10000).plot()

(W - 120 | (W > 120) ).sim(10000).plot()

plt.show();

Exponential race

X, Y = RV(Exponential(rate = 1 / 10) * Exponential(rate = 1 / 20))

W = (X & Y).apply(min)

xyw = (X & Y & W).sim(10000)
xyw

Index	Result
0	(4.983024727303675, 1.8686496510707014, 1.8686496510707014)
1	(3.532554503008326, 9.05367112944098, 3.532554503008326)
2	(24.918965681064886, 13.345421877996436, 13.345421877996436)
3	(1.0775484149017658, 42.42848811731041, 1.0775484149017658)
4	(8.548471948094639, 7.969724384024137, 7.969724384024137)
5	(2.799644404617912, 44.18113902851118, 2.799644404617912)
6	(13.52302122544879, 5.549623631722809, 5.549623631722809)
7	(2.222538632722782, 7.602869821226346, 2.222538632722782)
8	(3.3007495617180953, 8.126766296265734, 3.3007495617180953)
...	...
9999	(23.287665393006094, 0.410007076623495, 0.410007076623495)

xyw.mean()

(9.853554692021284, 20.148862821285917, 6.602680210076662)

plt.figure();

W.sim(10000).plot()

Exponential(rate = 1 / 10 + 1 / 20).plot()

plt.show();

(Y > X).sim(10000).tabulate(normalize = True)

Outcome	Relative Frequency
False	0.339
True	0.661
Total	1.0

plt.figure();

(W | (Y > X) ).sim(10000).plot()

(W | (Y < X) ).sim(10000).plot()

Exponential(rate = 1 / 10 + 1 / 20).plot()

plt.show();

Gamma distributions

W1, W2, W3 = RV(Exponential(rate = 2) ** 3)

T = W1 + W2 + W3

sim = (W1 & W2 & W3 & T).sim(10000)

sim

Index	Result
0	(0.4052504524799489, 0.5114739087659492, 0.2816990600116868, 1.198423421257585)
1	(0.2845006034749414, 0.784147170615129, 0.06580140653945067, 1.1344491806295212)
2	(0.29037566745592863, 0.40689436589149, 1.2309530778525988, 1.9282231112000174)
3	(0.3761683472390624, 0.3912870039018704, 1.7872597578528382, 2.554715108993771)
4	(0.601298208076717, 0.14219483464185464, 0.503822210308268, 1.2473152530268397)
5	(1.3643831264151047, 0.6027998436707337, 0.005046870394153958, 1.9722298404799925)
6	(2.131537555459863, 0.8582450938251214, 0.01889120059840637, 3.0086738498833903)
7	(0.004463730964075298, 0.12675742084063174, 0.2831797269746502, 0.4144008787793573)
8	(0.7406988438093932, 0.17534287199045778, 0.025007696913787885, 0.941049412713639)
...	...
9999	(0.4468714918036265, 1.2508675378247072, 0.4194708243947566, 2.1172098540230904)

sim.mean()

(0.4994192595452543, 0.5008799046109623, 0.5030317990526548, 1.5033309632088712)

sim.var()

(0.2519628669400152, 0.25448442216859424, 0.25792731168795424, 0.7653816655218177)

sim.sd()

(0.5019590291448249, 0.5044644904932301, 0.5078654464402498, 0.8748609406767557)

plt.figure();

T.sim(10000).plot()

Gamma(shape = 3, rate = 2).plot()

plt.show();