98 Multivariate Distributions2.1.2.LetA 1 ={(x, y):x≤ 2 ,y≤ 4 },A 2 ={(x, y):x≤ 2 ,y≤ 1 },A 3 =
{(x, y):x≤ 0 ,y≤ 4 },andA 4 ={(x, y):x≤ 0 y≤ 1 }be subsets of the
spaceAof two random variablesXandY, which is the entire two-dimensional
plane. IfP(A 1 )=^78 ,P(A 2 )=^48 ,P(A 3 )=^38 ,andP(A 4 )=^28 , findP(A 5 ), where
A 5 ={(x, y):0<x≤ 2 , 1 <y≤ 4 }.
2.1.3.LetF(x, y) be the distribution function ofXandY. For all real constants
a<b, c<d, show thatP(a<X≤b, c < Y≤d)=F(b, d)−F(b, c)−F(a, d)+
F(a, c).2.1.4.Show that the functionF(x, y) that is equal to 1 provided thatx+2y≥1,
and that is equal to zero provided thatx+2y<1, cannot be a distribution function
of two random variables.
Hint: Find four numbersa<b, c<d,sothat
F(b, d)−F(a, d)−F(b, c)+F(a, c)is less than zero.
2.1.5.Given that the nonnegative functiong(x) has the property that
∫∞0g(x)dx=1,show that
f(x 1 ,x 2 )=
2 g(√
x^21 +x^22 )
π√
x^21 +x^22, 0 <x 1 <∞, 0 <x 2 <∞,zero elsewhere, satisfies the conditions for a pdf of two continuous-type random
variablesX 1 andX 2.
Hint: Use polar coordinates.
2.1.6.Consider Example 2.1.3.
(a)Show thatP(a<X<b,c<Y <d)=(exp{−a^2 }−exp{−b^2 })(exp{−c^2 }−
exp{−d^2 }).(b)Using Part (a) and the notation in Example 2.1.3, show thatP[(X, Y)∈A]=
0 .1879 whileP[(X, Y)∈B]=0.0026.(c)Show that the following R program computesP(a<X<b,c<Y <d).
Then use it to compute the probabilities in Part (b).plifetime <- function(a,b,c,d)
{(exp(-a^2) - exp(-b^2))*(exp(-c^2) - exp(-d^2))}2.1.7.Letf(x, y)=e−x−y, 0 <x<∞, 0 <y<∞, zero elsewhere, be the pdf of
XandY.ThenifZ=X+Y, computeP(Z≤0),P(Z≤6), and, more generally,
P(Z≤z), for 0<z<∞. What is the pdf ofZ?