scipy.stats for all problems in this set.Your Markdown answer should look like the following:
Distribution: Bernoulli
p: 0.3
we are being asked to calculate the expected values, whose equation is:
$\sum_{x=0}^1 x P(x)$
Using the python cell below, the answer is $0.3$
The time between Siggi, my puppy, stealing socks is exponentially distributed. He steals a sock about every 6 hours. What's the probability that we go an entire day without him stealing a sock?
Distribution: Exponential
$\lambda$: 1 / 6 socks per hour
We are being asked a cumulative probability:
$$ \int_{24}^\infty \lambda e^{-\lambda x} \, dx $$which is 0.018.
In [18]:
import scipy.stats as ss
1 - ss.expon.cdf(24, scale=6)
Out[18]:
My cat Pizzicato loves to go outside and I let her out about once every other day. With a probability of 99%, what's the longest amount of time between Pizzicato going outside. Another way of saying it: what's the amount of time that corresponds to 99% of the probability between trips outside for Pizzicato
Distribution: Exponential
$\lambda$: 0.5 per day
We are being asked to compute a prediction interval for 99%:
$$ \int_0^x \lambda e^{-\lambda t}\,dt = 0.99 $$which is 9.2 days
In [19]:
ss.expon.ppf(0.99, scale=2)
Out[19]:
At a manufacturing plant, there are on average 2 defective parts per one thousand. What's the probability of less than 5 defective parts in a given lot of one thousand parts?
Distribution: Poisson
$\mu$: 2
We are being asked a cumulative distribution function
$$ \sum_{x=0}^4 \frac{e^{-\mu}\mu^x}{x!} $$which is 0.947
In [20]:
ss.poisson.cdf(4, 2)
Out[20]:
At a manufacturing plant, the quality assurance inspector examines five parts from a lot of one thousand and rejects the entire lot if any of them are defective. If we have the same rate of defects, 2 part per thousand, then what's the probability that a lot is rejected?
Distribution: Geometric
p: 2 / 1000
We are being asked about a cumulative distribution function:
$$ \sum_{x=1}^5 p(1-p)^{x - 1} $$which is 0.0100
In [21]:
ss.geom.cdf(5, 2 / 1000)
Out[21]:
A reaction vessel can tolerate a maximum temperature of 2250 $^\circ$C. You measure the temperature in the vessel many times and determine the average temperature is 1350 $^\circ$ C with a standard deviation of 400$^\circ$ C. What's the probability that the temperature in the reaction vessel exceeds its maximum temperature?
Distribution: Normal
$\mu$: 1350
$\sigma$: 400
We are being asked about this cumulative probability distribution interval:
$$ \int_{2250}^\infty {\mathcal{N}(1350, 400)}\,dx $$which has a probability of 0.0122
In [22]:
1 - ss.norm.cdf(2250, loc=1350, scale=400)
Out[22]:
English Mastiff dogs weigh 134 pounds on average with a standard deviation of 12 pounds. What's the probability of having a Mastiff that weighs less than 100 pounds?
Distribution: Normal
$\mu$: 134
$\sigma$: 12
We are being asked about this cumulative probability distribution interval:
$$ \int_{-\infty}^{100} {\mathcal{N}(134, 12)}\,dx $$which has a probability of 0.00230
In [23]:
ss.norm.cdf(100, scale=12, loc=134)
Out[23]:
In a class, the average score on a test is 94% with a standard deviation of 13%. Can we model this with a normal distribution? Why or why not? Assume that the highest possible score is 100% (no extra credit).
Distribution: Normal
$\mu$: 94
$\sigma$: 13
We are being asked about this cumulative probability distribution interval:
$$ \int_{100}^{\infty} {\mathcal{N}(134, 12)}\,dx $$which has a probability of 0.32. No, this is not a good approximation since the model predicts 32% of the students have greater than 100%.
In [24]:
1 - ss.norm.cdf(100, scale=13, loc=94)
Out[24]: