For this problem (and others related to normal distributions), you may want to use the norm functions in scipy.stats. If X is normally distributed with a mean of 6 and a variance of 25, then find

• (a) P(6≤X≤12)
• (b) P(0 ≤ X ≤ 8)
• (c) P(X > 21)
• (d) P(|X−6|<15)

13. For this problem, you will need the data in week6-4.txt.

• (a) Construct a frequency histogram for the data in the file.
• (b) Compute the mean xˉ and standard deviation s of the data.

14. For this problem, you will need the data in week6-5.txt.

• (a) Using NumPy, load the data into an array. Then find the mean and standard deviation of the data.
• (b) Using NumPy broadcasting, convert each value to its z-score. You do not need to write out all 200 z-scores. Instead, write out the first five of them.

15. Let the joint pmf of X and Y be given by

f(x,y)=32x+y,x=1,2,y=1,2,3,4

Find the following:

• (a) f₁(x), the marginal pmf of X
• (b) f2(x), the marginal pmf of Y
• (c) P (X > Y )
• (d) P(Y = 2X)
• (e) P (X ≤ 3 − Y )
• (f) Are X and Y independent? Explain your answer.

16. Roll a pair of four-sided dice, one red and one black. Let X equal the outcome on the red die and let Y equal the sum of the two dice.

• (a) Describe the sample space of X and Y .
• (b) Define the joint pmf. It might be helpful to create this pmf using points (x, y) in the plane where at the point (x,y), the value associated to the point is the probability P(X=x,Y =y).
• (c) Find the marginal pmf of X.
• (d) Find the marginal pmf of Y .
• (e) Are X and Y independent? Explain your answer.
• (f) Determine μ_X, μ_Y, σ_X² ,σ_Y² .
• (g) Determine the covariance of X and Y , Cov(X, Y ), and the correlation coefficient ρ.
• (h) Find the equation of the least squares regression line and draw it on your graph from part (b). Note: The least-squares regression line can be found as
  y=μY+ρ⋅σXσY(x−μX)

17. Let the joint pmf of X andY by

f(x,y)=41,(x,y)ϵ{(0,0),(1,1),(1,−1),(2,0)}

• (a) Are X and Y independent? Explain your answer.
• (b) Determine the covariance of X and Y , Cov(X, Y ), and the correlation coefficient ρ.
• (c) What does this problem tell you about dependent random variables and their correlation coefficient?

18. For this problem, you are encouraged to use Python and you will need the data in week6-6.txt. The data in the file gives the ACT Math and ACT Verbal scores x,y, respectively, for 15 students.

• (a) Find the mean of both the Math and Verbal scores, as well as the variance for both.
• (b) Find the sample correlation coefficient, r.
• (c) Find the equation of the least squares regression line for the data
• (d) Plot the points and the equation on the same graph.

19. The joint pmf of X andY is 

f(x,y)=61, 0≤x+y≤2

where x and y are non-negative integers.

• (a) Sketch the domain of f.
• (b) Determine the marginal pmf's of X and Y .
• (c) Find E[Y|x] (i.e. the conditional mean of Y given X = x) and plot the points (x, E[Y |x]) on the sketch from part (a).
• (d) Find E[X|y] and plot the points (E[X|y],y) on the sketch from part (a).
• (e) Find ρ using your answers from parts (c) and (d).

20. Let X and Y have a bivariate normal distribution with parameters μ_X = −3, μ_Y = 10,  σ²_X =25, σ²_Y=9, and ρ=53. Compute

• (a) P(−5 < X < 5)
• (b) P(−5 < X < 5|Y = 13)
• (c) P(7 < Y < 16)
• (d) P(7 < Y < 16|X = 2)

21. Let X and Y have a bivariate normal distribution with parameters μ_X = 70, μ_Y = 80,  σ²_X = 100, σ²_Y= 169, and ρ = 53 . Compute

• (a) E[Y |X = 72] (i.e. the conditional mean of Y given X = 72)
• (b) Var(Y |X = 72) (i.e. the conditional variance of Y , given X = 72)
• (c) P(Y ≤ 84|X = 72)

 

week6-4.txt

33.11418075586503
35.232897745035615
34.74166062148456
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31.701907433932714
39.53191596488375
38.062732231195525
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week6-5.txt

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101.76425985212984

week6-6.txt

16,19
18,17
22,18
20,23
17,20
25,21
21,24
23,18
24,18
31,25
27,29
28,24
36,29
27,23
28,24