Description

Problem 2

Part A

Out[3]: 0.06680720126885807

Out[4]: 0.5793625654038657

Out[5]: 0.35383023332727626
Part B
In [6]: lis = gauss.rvs(1000000) greater_than_80 = lis >= 80 greater_than_85 = lis >= 85 np.sum(greater_than_85)/np.sum(greater_than_80)
Out[6]: 0.5911556069997511
Part C
In [7]: def gauss_pdf(mu, var, x):
return 1/(2*np.pi*var)**(1/2)*np.exp(-(x-mu)**2/(2*var))
In [8]: for mu in range(30,100): for var in range(30,200):
prob_a = 0 prob_b = 0
for x_a in range(85,100): # calculate the cdf of a prob_a+=gauss_pdf(mu,var,x_a)
for x_b in range(70,85): # calculate the cdf of b prob_b+=gauss_pdf(mu,var,x_b) if (abs(prob_a-.25) <= .0025 and (abs(prob_b-.52) <= .0025)): print(“Mu = “,mu,”. Var = “,var,”. Prob of A = “,prob_a,”. P rob of B = “,prob_b,”.”,sep=””)
Mu = 78. Var = 111. Prob of A = 0.2479537739631005. Prob of B = 0.52163
53936665281.
Mu = 78. Var = 112. Prob of A = 0.24840772370227115. Prob of B = 0.5196
808507166172.
Mu = 78. Var = 113. Prob of A = 0.24884873008939257. Prob of B = 0.5177 478397299403.
In [9]: new_gauss = stats.norm(loc=78, scale = 112**.5) # create a new gaussian with the paramaters found above
In [10]: new_gauss.cdf(85)-new_gauss.cdf(70) # probability of a B
Out[10]: 0.5209883142394818

Out[11]: 0.2541657867760727
Problem 5
a. Define the pmf function of the binomial random variable
In [13]: sum = 0
b. Random variable – summing the pmf

In [15]: sum = 0
Out[15]: 0.7019577122638231
c. Random variable – using the cdf

Out[16]: 0.7039868846425189
Problem 6
Part A

MAP decision boundary:
Choose the Power Amplifier for any point with an x less than 1.35955598
68917453,0.07916017444797833 or an x greater than -1.3595559868917453, 0.07916017444797833. For the region inbetween these two points, choose the Low Noise Amplifier.
Part B

Decide Power Amplifier, given x=-7.5
Decide Power Amplifier, given x=-7.3
Decide Power Amplifier, given x=-7.1
Decide Power Amplifier, given x=-6.9
Decide Power Amplifier, given x=-6.7
Decide Power Amplifier, given x=-6.5
Decide Power Amplifier, given x=-6.3
Decide Power Amplifier, given x=-6.1
Decide Power Amplifier, given x=-5.9
Decide Power Amplifier, given x=-5.7
Decide Power Amplifier, given x=-5.5
Decide Power Amplifier, given x=-5.3
Decide Power Amplifier, given x=-5.1
Decide Power Amplifier, given x=-4.9
Decide Power Amplifier, given x=-4.7
Decide Power Amplifier, given x=-4.5
Decide Power Amplifier, given x=-4.3
Decide Power Amplifier, given x=-4.1
Decide Power Amplifier, given x=-3.9
Decide Power Amplifier, given x=-3.7
Decide Power Amplifier, given x=-3.5
Decide Power Amplifier, given x=-3.3
Decide Power Amplifier, given x=-3.1
Decide Power Amplifier, given x=-2.9
Decide Power Amplifier, given x=-2.7
Decide Power Amplifier, given x=-2.5
Decide Power Amplifier, given x=-2.3
Decide Power Amplifier, given x=-2.1
Decide Power Amplifier, given x=-1.9
Decide Power Amplifier, given x=-1.7
Decide Power Amplifier, given x=-1.5
Decide Low Noise Amplifier, given x=-1.3
Decide Low Noise Amplifier, given x=-1.1
Decide Low Noise Amplifier, given x=-0.9
Decide Low Noise Amplifier, given x=-0.7
Decide Low Noise Amplifier, given x=-0.5
Decide Low Noise Amplifier, given x=-0.3
Decide Low Noise Amplifier, given x=-0.1
Decide Low Noise Amplifier, given x=0.1
Decide Low Noise Amplifier, given x=0.3
Decide Low Noise Amplifier, given x=0.5
Decide Low Noise Amplifier, given x=0.7
Decide Low Noise Amplifier, given x=0.9
Decide Low Noise Amplifier, given x=1.1
Decide Low Noise Amplifier, given x=1.3
Decide Power Amplifier, given x=1.5
Decide Power Amplifier, given x=1.7
Decide Power Amplifier, given x=1.9
Decide Power Amplifier, given x=2.1
Decide Power Amplifier, given x=2.3
Decide Power Amplifier, given x=2.5
Decide Power Amplifier, given x=2.7
Decide Power Amplifier, given x=2.9
Decide Power Amplifier, given x=3.1
Decide Power Amplifier, given x=3.3
Decide Power Amplifier, given x=3.5
Decide Power Amplifier, given x=3.7 Decide Power Amplifier, given x=3.9
Decide Power Amplifier, given x=4.1
Decide Power Amplifier, given x=4.3
Decide Power Amplifier, given x=4.5
Decide Power Amplifier, given x=4.7
Decide Power Amplifier, given x=4.9
Decide Power Amplifier, given x=5.1
Decide Power Amplifier, given x=5.3
Decide Power Amplifier, given x=5.5
Decide Power Amplifier, given x=5.7
Decide Power Amplifier, given x=5.9
Decide Power Amplifier, given x=6.1
Decide Power Amplifier, given x=6.3
Decide Power Amplifier, given x=6.5
Decide Power Amplifier, given x=6.7
Decide Power Amplifier, given x=6.9
Decide Power Amplifier, given x=7.1
Decide Power Amplifier, given x=7.3
Part C

Part D

Part E

Part F

Part G

When the probability of a random chosen chip being a low noise amplifier is 0.2, the math breaks down and we always decide that it’s a power amplifier and never decide that it’s a low noise amplifier.
Part H

Part I
In [29]: for x_int in range (40,20,-1): x = x_int/100
print(“Probability of a randomly chosen chip being a low-noise ampli fier=”,x,”.”,sep=””) vary_densities(x)
Probability of a randomly chosen chip being a low-noise amplifier=0.4.

Probability of a randomly chosen chip being a low-noise amplifier=0.39.

Probability of a randomly chosen chip being a low-noise amplifier=0.38.

Probability of a randomly chosen chip being a low-noise amplifier=0.37.

Probability of a randomly chosen chip being a low-noise amplifier=0.36.

Probability of a randomly chosen chip being a low-noise amplifier=0.35.

Probability of a randomly chosen chip being a low-noise amplifier=0.34.

Probability of a randomly chosen chip being a low-noise amplifier=0.33.

Probability of a randomly chosen chip being a low-noise amplifier=0.32.

Probability of a randomly chosen chip being a low-noise amplifier=0.31.

Probability of a randomly chosen chip being a low-noise amplifier=0.3.

Probability of a randomly chosen chip being a low-noise amplifier=0.29.

Probability of a randomly chosen chip being a low-noise amplifier=0.28.

Probability of a randomly chosen chip being a low-noise amplifier=0.27.

Probability of a randomly chosen chip being a low-noise amplifier=0.26.

Probability of a randomly chosen chip being a low-noise amplifier=0.25.

Probability of a randomly chosen chip being a low-noise amplifier=0.24.

Probability of a randomly chosen chip being a low-noise amplifier=0.23.

Probability of a randomly chosen chip being a low-noise amplifier=0.22.

Probability of a randomly chosen chip being a low-noise amplifier=0.21.

As seen above, the logic breaks down when the probability of a randomly chosen chip being a low-noise amplifier is approximately one third.