Final answer:
To find the probability that one sample observation is at least twice as large as the other, we need to determine the joint pdf of the two random variables and integrate it over the appropriate region.
Step-by-step explanation:
To find the probability that one sample observation is at least twice as large as the other, we need to determine the joint pdf of the two random variables. Let's define X as the smaller sample observation and Y as the larger sample observation. Since these are independent and identically distributed random variables, we can use the joint pdf formula: fXY(x,y) = fX(x) * fY(y).
Since the pdf of X is given as fX(x) = 2(1-x), 0 < x < 1, we can substitute this into the joint pdf formula to get: fXY(x,y) = 2(1-x) * 2(1-y).
To find the probability that one sample observation is at least twice as large as the other, we need to integrate the joint pdf over the appropriate region. In this case, the region is defined by the condition y >= 2x. So the probability can be calculated as:
P(Y >= 2X) = ∫2x1 ∫y1 fXY(x,y) dy dx