# Econ7630_Exam1a

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1. PDF Facts
﻿﻿P[X = x] >= 0

Sum[f(x)] = 1
2. Random Variable
A function that assigns a unique numerical value to each sample space outcome.
3. CDF
﻿﻿
4. Given this discrete CDF, what is P[X<=1.5]?

x f(x)

1 0.1
2 0.2
3 0.3
4 0.4
0.1
5. Joint PDF: f (X, Y) for discrete r.v.
P[X = x, Y = y]
6. Joint PDF: f (X, Y) for continous r.v. over a certain range.
7. Marginal PDF: f (x) for discrete r.v.
f(x) = sumy[f(x,y)]

f(y) = sumx[f(x,y)]
8. ﻿﻿Marginal PDF: f (X) for continous r.v.
9. Conditional Probability: P[X = x | Y = y] (i.e. f(X|Y))
= P[X = x, Y = y]/P[Y = y] = f(x, y)/f(y)

The conditional probabilty that X equals x given that Y equals y is equal to the joint PDF of X and Y divided by the marginal PDF of y.
10. X, Y are independent IFF
(1) f(x, y) = f(x)f(y) for all x, y.

• Or (derived from (1))
• (2) f(x|y) = f(x),
• (3) f(y|x) = f(y) for all x, y.
11. Given that X, Y are independent show that f(x|y) = f(x) for all x, y.
• We know that
• (i) X,Y are independent if f(x,y) = f(x)f(y) for all x,y.
• (ii) f(x|y) = f(x,y)/f(y)

So by (i) f(x|y) = f(x)f(y)/f(y) -> f(x|y) = f(x).
12. Expected Value: E[X] for
(1) Discrete
(2) Continous
13. What does Expected Value mean for a discrete random variable?
The average value of X on an infinite number of experimental trials.

For example, E[X] for a six sided fair die is 3.5. Clearly saying something like the likely value of a die role is 3.5 does not make sense. Average makes sense.
14. Let g(X) be a function of X. What is E[g(X)]?
• ﻿
• Remember: even though X is being transformed by g(X), you still use the original PDF f(x).
15. If g(X) equals c, a constant, then E[g(X)] =
E[c] = c

Remember: The Expected Value of a constant is that constant. This allows you to pull constants out of the E[ ] operator.
16. Show that E[c] = c, where c is a constant.
Let g(X) = c -> E[g(X)] = integral[g(x)f(x)dx] = integral[cf(x)dx] = c*integral[f(x)dx] = c(1) = c
17. If g(X) = aX + b, then E[g(X)] =
E[aX + b] = aE[X] + b
18. Show that E[aX + b] = aE[X] + b
Let g(X) = aX + b -> E[g(X)] = E[aX + b] = E[aX] + E[b] -> aE[X] + b
19. Let g(X) = g1(X) + g2(X) + . . . + gn(X),

what is E[g(X)]?
• Remember: the expected value of the sum is the sum of the expected values.
20. Var[X]
= E[(X - mu)2] = E[X2] - mu2, where mu = E[X].

• Proof:
• E[(X - mu)2] = E[(X - mu)(X - mu)] = E[X2 - 2muX + mu2] = E[X2] - 2muE[X] + mu2 = E[X2] - 2mu*mu + mu2 = E[X2] - 2mu2 + mu2 = E[X2] - mu2
21. Let Y = a + bX, what is Var[Y]?
= b2Var[X]

• Proof:
• Let Y = a + bX -> E[Y] = a + bE[Y]. Then Var(Y) =
22. Standardized Variable:

z =
• (x - mu)/sig
• where sig = sqrt[Var(X)] (i.e. the standard deviation)
23. Show that E[z] = 0
• ﻿E[z] = E[(1/sig)X - mu/sig]
• = (1/sig)E[X] - E[mu/sig]
• = mu/sig - mu/sig = 0
24. Show that Var(z) = 1
• ﻿Var(z) = Var[(1/sig)X - mu/sig]
• = (1/sig)2Var(X) = sig2/sig2 = 1

Remember: to get to line two, Var(aX +- b) = a2Var(X).
25. Let g(X1, X2) have joint PDF f(X1, X2).

E[g(X1, X2)] =
﻿
26. cov(X1, X2)
• ﻿= E[X1, X2] - mu1mu2
• Derivation:
• ﻿
27. If cov(X,Y)
1. > 0
2. < 0
3. = 0
• 1. (X, Y) pairs tend to be both greater than their means or both less than their means.
• 2. (X, Y) pairs tend to be mixed about their means (one greater and one less)
28. Correlation
﻿
29. If p (for correleation)
1. = 1
2. = -1
3. = 0
• 1. perfect positive relation
• 2. perfect negative relation
• 3. no linear relationship

Remember: absolute_value(p) measure the strength of the linear relationship.
30. When X1 and X2 are independent, E[X1, X2]
= mu1mu2

• Proof:
31. If X, Y are independent then cov(X, Y)
= 0

Remember: the converse is not true.

• Proof:
• We know E[X, Y] = muXmuY, when X, Y ind.

cov[X, Y] = E[X, Y]-muXmuY = muXmuY-muXmuY = 0
32. Let c, d be constants

E[cX + dY]
= cE[X] + dE[Y] = c*muX + d*muY
33. Var(c1X1 + c2X2)
• c12Var(X1) + c22Var(X2) + 2c1c2Cov(X1, X2)
• Proof:
• Var(c1X1 + C2X2)
34. E[X|Y=y] for continous
• = integral(x*f(x|y)dx)
• Note: f(x|y) = f(x,y)/f(y)
35. E[a + bX | X]
= a + b*mu
36. E[g(X)|X]
= E[g(X)]
37. Law of Iterated Expectations

E[Y]
• = Ex[Ey(Y|X)]
• Proof:
38. If E[Y|X] = E[Y] then cov(X,Y)
• = 0
• Proof:
• ﻿
39. Var(X|Y)
= E[X2|Y] - E[X|Y]2
40. Normal Distribution
41. Standard Normal Distribution

Z~(0,1)
﻿
42. If X~N(mu,sig2) and Y=aX + b, then Y~
Y~N((a*mu +b), a2sig2)﻿
43. If X,Y are normal then cov(X,Y)
cov(X,Y) = 0 <=> X,Y are independent.
 Author: mattstam ID: 65858 Card Set: Econ7630_Exam1a Updated: 2011-02-13 18:32:12 Tags: econometrics probability Folders: Description: Exam 1 (cards part a) for graduate level Econometrics 1 at LSU for Spring 2011. Includes many not too terribly rigorous "proofs." Show Answers: