Additional notes EC220
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1. The variance of a random variable X is defined as the expected value of (X - E(X))^2. The covariance between two random variables X and Y is defined as the expected value of (X - E(X))(Y - E(Y)). 2. Several limits of probabilities (plim rules) are derived from Slutsky's theorem. These include: the plim of a sum of random variables equals the sum of their individual plims; the plim of a constant multiplied by a random variable is that constant multiplied by the random variable's plim; and the plim of a function of a random variable is equal to that function evaluated at the random variable's plim.
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![1 Variance and covariance rules
V ar(X) = E[(X − E(X))2 ] = E[X 2 + E[X]2 − 2XE[X]] (1)
2 2 2 2 2
= E[X ] + E[X] − 2E[X] = E[X ] − E[X] (2)
⇒ E[X 2 ] = V ar(X) + E[X]2
Cov(X, Y ) = E[(X − E[X])(Y − E[Y ])] = (3)
= E[XY − XE[Y ] − E[X]Y + E[X]E[Y ]] (4)
= E[XY ] − E[X]E[Y ] (5)
⇒ E[XY ] = Cov(X, Y ) + E[X]E[Y ]
2 plim rules
These results are derived from Slutsky’s Theorem. See Dougherty (p. 33) for the exact
requirements. X, Y , Z are random variables, b is a constant and f (·) is a continuous
function.
plim(X + Y + Z) = plim(X) + plim(Y ) + plim(Z) (6)
plim(b × X) = b × plim(X) (7)
plim(b) = b (8)
plim(X × Y ) = plim(X) × plim(Y ) (9)
plim(X/Y ) = plim(X)/plim(Y ) (10)
plimf (X) = f (plim(X)) (11)
Furthermore, these results may be assumed (according to past exam description):
1 N
plim Σ xi = E[X] = µx (12)
N i=1
1 N
plim Σ (xi − x)2 = V ar(X) = σx
¯ 2
(13)
N i=1
1 N 2
plim Σ (xi − x)(yi − y ) = Cov(X, Y ) = σxy
¯ ¯ (14)
N i=1
Aside: To obtain these results, one would need to invoke the appropriate law of large
numbers.
1](https://image.slidesharecdn.com/noteseconometrics-130212063936-phpapp01/85/Additional-notes-EC220-1-320.jpg)
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The document discusses methods for solving first order ordinary differential equations (ODEs). It covers:
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2) Solving homogeneous first order linear ODEs by making a substitution to reduce it to a separable equation.
3) Solving inhomogeneous first order linear ODEs using an integrating factor.
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Additional notes EC220
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