For the following problems, choose the distribution that best fits the process and plot the distribution. State whether the sample space is discrete, the range of the sample space, and then use that to justify your choice of distribution. Make sure your axes labels are descriptive and not just the variable letters.
In [33]:
import matplotlib.pyplot as plt
import numpy as np
from scipy.special import comb, factorial
n = np.arange(10)
mu = 3
p = mu**n * np.exp(-mu) / factorial(n)
plt.xlabel('Number of Deer [$n$]')
plt.ylabel('P(n)')
plt.plot(n, p, '-o')
plt.show()
In [34]:
from math import pi
mu = 8
s = 2
x = np.linspace(2, 14, 1000)
Px = 1 / np.sqrt(2 * pi) / s * np.exp(-(mu - x)**2 / (2 * s))
plt.plot(x, Px)
plt.xlabel('Hours Slept [$t$]')
plt.ylabel('P(t)')
plt.show()
You are going out to read your poetry. You can go to only one of three different poetry readings. At each poetry reading, they randomly select a poet to read for a given number of timeslots. You can read multiple times, since you have many poems to read. At CLUB POET, there will be 11 poem reading timeslots and your probability of reading at particular timeslot is 0.4. At THE DEAD HEDGEHOG, there will be 5 timeslots and your probability of reading will be 0.5. At THE DANCING CLOUD, there are 7 timeslots and your probability of reading will be 0.7. Plot each of the distributions representing number of poems read for the different poetry readings. You should plot three distributions.
Each have a maximum number of poems read, which is the discrete upper limit for sample space. 0 is the lower bound. Binomial best fits.
In [40]:
Ns = [11, 5, 7]
ps = [0.4, 0.5, 0.7]
names = ['Club Poet', 'The Dead Hedgehog', 'The Dancing Cloud']
for N,p, l in zip(Ns, ps, names):
n = np.arange(N + 1)
plt.plot(n, comb(N, n) * p**n * (1 - p)**(N - n), '-o', label=l)
plt.legend()
plt.xlabel('Poems Read [$n$]')
plt.ylabel('$P(n)$')
plt.show()
$t > 0$, continuous, exponential best fits.
In [43]:
t = np.linspace(0,60, 1000)
lamb = 1 / 20.
plt.plot(t, lamb * np.exp(-lamb * t))
plt.xlabel('time since last message [$t$]')
plt.ylabel('$P(t)$')
plt.show()
For the following problems, use loops to solve them. You cannot use numpy arrays. Justify your choice of distribution, state what quantity is being asked (probability of sample, interval, or prediction interval), and answer the prompt in words. The only formula you may use for expected value is $E[x] = \sum P(x) x$.
In [51]:
N = 12
p = 0.7
psum = 0
for ni in range(N + 1):
psum += comb(N, ni) * p**ni * (1 - p)**(N - ni)
if psum > 0.9:
break
print(ni)
In [8]:
p = 25 / 1e5
psum = 0
i = 1
while psum < 0.1:
psum += p*(1 - p)**i
i += 1
print(i)
In a computer program, each line of code can cause a security vulnerability. A particular company finds that after 1 year, there are 25 security vulnerabilities per 10,000 lines of code. After inspecting a new program which contains 23,000 lines of code, they find 11 secutity vulnerabilities. What is the probability that they found all vulnerabilities?
In [56]:
psum = 0
mu = 25
for i in range(12):
psum += mu**i * np.exp(-mu) / factorial(i)
print(psum)
In [57]:
ss.norm.cdf(0, scale=4, loc=3) - ss.norm.cdf(-4, scale=4, loc=3)
Out[57]:
In [58]:
1 - ss.norm.cdf(120, loc=111, scale=30)
Out[58]:
In [60]:
ss.norm.cdf(0, loc=111, scale=30)
Out[60]:
In [61]:
(-4 -- 2)/4, (4 -- 2) / 4
Out[61]:
In [62]:
ss.norm.cdf(3) - ss.norm.cdf(-3)
Out[62]: