Question

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

Answer 1

Answer:

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%)