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) = 4xy, 0

the probability that 0≤x≤1/2 and 1/4≤y≤1/2 is 3/64 (4.68%)

Step-by-step explanation:

assuming that X and Y are independent variables for the probability density function f (x, y):

f (x, y) = 4*x*y for 0≤x≤1 and 0≤y≤0

f (x, y) = 0 elsewhere

then the probability is calculated through:

P (x, y) = ∫f (x, y) dx dy = ∫4xy dx dy

for 0≤x≤1/2 and 1/4≤y≤1/2 we have

P (0≤x≤1/2,1/4≤y≤1/2) = ∫4xy dx dy = ∫4xy dx dy = 4*∫x dx ∫y dx = 4*[ ((1/2) ²/2-0²/2) ] * [ (1/2) ²/2 - (1/4) ²/2) ] = 1 * 1/4 * (1/4-1/16) = 1 * 1/4 * 3/16 = 3/64

then the probability that 0≤x≤1/2 and 1/4≤y≤1/2 is 3/64 (4.68%)