By Edwin Bidwell Wilson

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**Read or Download Advanced calculus; a text upon select parts of differential calculus, differential equations, integral calculus, theory of functions; with numerous exercises PDF**

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**Additional resources for Advanced calculus; a text upon select parts of differential calculus, differential equations, integral calculus, theory of functions; with numerous exercises **

**Sample text**

5) The means are E {Y1 } = E {X1 + X2 } = E {X1 } + E {X2 } = 5 3 +1= . 2 2 Alternatively, E {Y1 } = ∞ ∞ y1 kY1 (y1 ) dy1 = 0 0 5 6 2 1 3 1 2 −y1 y + y e dy1 = + = . com 62 5 3 1 − = . 4 3 12 4. 11 Let (X, Y ) be a 2-dimensional random variable of the frequency ⎧ −y for 0 ≤ x ≤ y, ⎨ e , h(x, y) = ⎩ 0, otherwise. 1) Find the marginal frequencies of the random variables X and Y . 2) Find the means E{X} and E{Y } of the random variables X and Y . 3) Find the variances V {X} and V {Y } of the random variables X and Y .

2) Clearly, fX (x) = 0 for x ∈ / ]0, 1[. When x ∈ ]0, 1[ it follows by a vertical integration that 1 fX (x) = 1 y2 2 8xy dy = 8x x x = 4x − 4x3 , hence the marginal frequency is ⎧ ⎨ 4x − 4x3 , x ∈ ]0, 1[, fX (x) = ⎩ 0, otherwise. com 39 4. Frequencies and distributions functions, 2 dimensions Random variables I thus the marginal distribution function is ⎧ 0, x ≤ 0, ⎪ ⎪ ⎪ ⎪ ⎨ 2x2 − x4 , 0 < x < 1, FX (x) = ⎪ ⎪ ⎪ ⎪ ⎩ 1, x ≥ 1. Clearly, fY (y) = 0 for y ∈ / ]0, 1[. If y ∈ ]0, 1[, we get by a horizontal integration that y fY (y) = x2 2 8xy dx = 8y 0 y = 4y 3 , 0 and the marginal frequency is ⎧ ⎨ 4y 3 , y ∈ ]0, 1[, fY (y) = ⎩ 0, otherwise.

0, 5) Medians: 1 1 = 1 − e−x for e−x = , hence x = (X) = ln 2. 2 2 1 1 1 1 for = , thus y = (Y ) = 1. b) Fy (y) = = 1 − 2 y+1 y+1 2 a) FX (x) = 6) Since fX (x) · fY (y) = e−x = x e−x(y+1) = f (x, y) (y + 1)2 for x, y > 0, X and Y are not independent. 4 A 2-dimensional random variable (X, Y ) has the frequency ⎧ ⎨ c x y, 0 < x < y < 1, f (x, y) = ⎩ 0, otherwise. Find the constant c. Find the frequencies and the distribution function of the random variables X and Y . Check if the random variables X and Y are independent.