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Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density f(x, y) = {?2 +??/3 ,0 < ? < 1, 0 < ? < 2, 0 elsewhere. a. Verify if it is a valid density function. b. Find P[(X, Y) ∈ A], where A = {(x, y) | 0 < x < ½, ¼ < y < ½. c. Find the marginal density g(x).

Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density f(x, y) = {?2 +??/3 ,0 < ? < 1, 0 < ? < 2, 0 elsewhere. a. Verify if it is a valid density function. b. Find P[(X, Y) ∈ A], where A = {(x, y) | 0 < x < ½, ¼ < y < ½. c. Find the marginal density g(x).

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.3: Special Probability Density Functions
Problem 6E
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Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F) at which a certain reaction starts to take place. Suppose that two 
random variables X and Y have the joint density
f(x, y) = {?2 +??/3 ,0 < ? < 1, 0 < ? < 2,
               0 elsewhere.
a. Verify if it is a valid density function.
b. Find P[(X, Y) ∈ A], where A = {(x, y) | 0 < x < ½, ¼ < y < ½.
c. Find the marginal density g(x). 
d. Find the marginal density h(y).
e. Find the conditional density f (x|y). 
f. Find the conditional density f (y|x). 
g. Find P(Y > 1 | X = ½). 
h. Find P(X > ½ | Y = 1). 
i. Determine if the random variables are statistically independent. 

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