9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y) = ㅠ for -∞ < x <∞ and -x < y < x; see also Figure 9.2. 30 -502²-50y² +80xy 134 9 Joint distributions and independence a. Determine positive numbers a, b, and c such that 50x² 80xy +50y² = (ay-bx)² + cr². b. Setting μ = r, and o= = 10, show that (√50y – √32x)² = ("−¹)² - 2 and use this to show that [ e-(√50y-√32x)² dy= = 2T 10 c. Use the results from b to determine the probability density function fx of X. What kind of distribution does X have?
9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y) = ㅠ for -∞ < x <∞ and -x < y < x; see also Figure 9.2. 30 -502²-50y² +80xy 134 9 Joint distributions and independence a. Determine positive numbers a, b, and c such that 50x² 80xy +50y² = (ay-bx)² + cr². b. Setting μ = r, and o= = 10, show that (√50y – √32x)² = ("−¹)² - 2 and use this to show that [ e-(√50y-√32x)² dy= = 2T 10 c. Use the results from b to determine the probability density function fx of X. What kind of distribution does X have?
9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y) = ㅠ for -∞ < x <∞ and -x < y < x; see also Figure 9.2. 30 -502²-50y² +80xy 134 9 Joint distributions and independence a. Determine positive numbers a, b, and c such that 50x² 80xy +50y² = (ay-bx)² + cr². b. Setting μ = r, and o= = 10, show that (√50y – √32x)² = ("−¹)² - 2 and use this to show that [ e-(√50y-√32x)² dy= = 2T 10 c. Use the results from b to determine the probability density function fx of X. What kind of distribution does X have?
9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for -
Transcribed Image Text:9.19 Let X and Y be two continuous random variables, with joint proba-
bility density function
f(x, y) =
30-502²-50y²+80zy
ㅠ
for -∞ < x < ∞ and - < y < x; see also Figure 9.2.
134 9 Joint distributions and independence
a. Determine positive numbers a, b, and c such that
50x² 80xy +50y² = (ay-bx)² + cx².
b. Setting u = , and olo, show that
=
and use this to show that
(√50y-√32.x)² -1 (³-7") ²
2
Le e-(√50y-√32x)² dy
=
√2π
10
c. Use the results from b to determine the probability density function fx
of X. What kind of distribution does X have?
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
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