9367 #40

[origin415]

If x, y, and z are selected independently and at random from the interval [0, 1], then the probability that x >= yz is
A) 3/4
B) 2/3
C) 1/2
D) 1/3
E) 1/4

The answer is A
I got it right, but couldn't figure out how to actually calculate it, I just guessed that it must be pretty high. Its obviously greater than 1/2, but my limited knowledge of probability theory prevented me from getting any further.

Thanks.

[mtey]

the triple density function

$$f_{XYZ}(x,y,z) = 1$$ since they are independent. So the probability is

$$\displaystyle \int \int \int_{D}1dxdydz$$ where
$$D = \begin{cases}
yz \leq x \leq 1 \\ 0 \leq y \leq 1 \\ 0 \leq z \leq 1 \\ \end{cases}$$

[origin415]

Ah, I was trying to do an integral, 0 < x, y < 1, 0 < z < x/y, which had quite a bit of complications with y going to zero and all. The other direction makes a lot more sense, thanks.

[yoyobarn]

the triple density function

$$f_{XYZ}(x,y,z) = 1$$ since they are independent. So the probability is

$$\displaystyle \int \int \int_{D}1dxdydz$$ where
$$D = \begin{cases}
yz \leq x \leq 1 \\ 0 \leq y \leq 1 \\ 0 \leq z \leq 1 \\ \end{cases}$$

Can someone elaborate more on this?
Is there a way without doing triple integration?

Thanks a lot.

[goombayao]

E(yz) = E(y)*E(z) = 1/2 * 1/2 = 1/4

P(x > 1/4) = 3/4

[yoyostein]

E(yz) = E(y)*E(z) = 1/2 * 1/2 = 1/4

P(x > 1/4) = 3/4

Can anyone explain why the above method works?

If I consider the expectation of X instead,
E(X)=1/2

P(yz > 1/2) != 1/2 ?


Sincere thanks for the clarification.

[Vince]

E(yz) = E(y)*E(z) = 1/2 * 1/2 = 1/4

P(x > 1/4) = 3/4

Can anyone explain why the above method works?

If I consider the expectation of X instead,
E(X)=1/2

P(yz > 1/2) != 1/2 ?


Sincere thanks for the clarification.

Since y and z are independent, so the mean of a product is a product of their means. Therefore, E(zy) = E(z)*E(y)=1/2*1/2=1/4.
In order for x > yz => x should be > 1/4. Again, x is chosen randomly(!), so we can figure out this probability easily: (1-1/4)/(1-0)=3/4.

Your method is incorrect, because using this solution you give an answer to another question.