w******g
发帖数: 67 | 1 A,C are iid random numbers uniformly distributed in (0,1)
B is random number uniformly distributed in (1/2,1)
What is the probability P(A>B,A>C)?
My answer is (1/2)*(3/8)=3/16. Is it correct? Thanks. | R******m
发帖数: 44 | 2 P(A>C)=1/2;
p(A>B)=1/2*1/2;
P(A>B,A>C)=P(A>C)*p(A>B)=1/8;
not sure ;
is that correct? | y***d
发帖数: 2330 | 3 no
【在 R******m 的大作中提到】
: P(A>C)=1/2;
: p(A>B)=1/2*1/2;
: P(A>B,A>C)=P(A>C)*p(A>B)=1/8;
: not sure ;
: is that correct?
| t****a
发帖数: 1212 | 4 It is easy to know that
D(A|A>C)=2A and P(A>C)=0.5 (Here D means prob density)
D(B) = 2 when 0.5
So D(A, B|A>C) = 2A * 2 (0.5 < B <1, 0.5 < A < 1)
P(A>B|A>C) = 定积分(0.5, 1)D(A, B|A>C)*(A-0.5)dAdB = 5/12
so P(A>B, A>C) = P(A>B|A>C) * P(A>C) = 5/24 | y***d
发帖数: 2330 | 5 D(A|A>C)=A 吧？
【在 t****a 的大作中提到】
: It is easy to know that
: D(A|A>C)=2A and P(A>C)=0.5 (Here D means prob density)
: D(B) = 2 when 0.5: So D(A, B|A>C) = 2A * 2 (0.5 < B <1, 0.5 < A < 1)
: P(A>B|A>C) = 定积分(0.5, 1)D(A, B|A>C)*(A-0.5)dAdB = 5/12
: so P(A>B, A>C) = P(A>B|A>C) * P(A>C) = 5/24
| d********t
发帖数: 9628 | 6
0.5*0.5*0.5 = 1/8?
【在 w******g 的大作中提到】
: A,C are iid random numbers uniformly distributed in (0,1)
: B is random number uniformly distributed in (1/2,1)
: What is the probability P(A>B,A>C)?
: My answer is (1/2)*(3/8)=3/16. Is it correct? Thanks.
| s****a
发帖数: 528 | 7 you are right.
【在 t****a 的大作中提到】
: It is easy to know that
: D(A|A>C)=2A and P(A>C)=0.5 (Here D means prob density)
: D(B) = 2 when 0.5: So D(A, B|A>C) = 2A * 2 (0.5 < B <1, 0.5 < A < 1)
: P(A>B|A>C) = 定积分(0.5, 1)D(A, B|A>C)*(A-0.5)dAdB = 5/12
: so P(A>B, A>C) = P(A>B|A>C) * P(A>C) = 5/24
| q**r
发帖数: 611 | | s****a
发帖数: 528 | 9 D(A,B,C) = 2 density function:
P(C<=0.5) = 1/2
P(A>B, A>C) = P (A>B) - P(A>B,C>=A)
= int B (0.5,1) int A(B,1) * P(A,B)
- int B(0.5,1) int A(B,1) int C(A,1) P(A,B,C)
= 5/24 | s****a
发帖数: 528 | 10 D(A,B,C) = 2 density function:
P(C<=0.5) = 1/2
P(A>B, A>C) = P (A>B) - P(A>B,C>=A)
= int B (0.5,1) int A(B,1) * P(A,B)
- int B(0.5,1) int A(B,1) int C(A,1) P(A,B,C)
= 5/24 | | | t****a
发帖数: 1212 | 11 注意条件概率密度D(A|A>C)=A在[0,1]间的积分仍然为1，所以应该是2A。
【在 y***d 的大作中提到】
: D(A|A>C)=A 吧？
| h**6
发帖数: 4160 | 12 纯直观的解：
假设A>1/2，C>1/2，概率分别为1/2，此时A、B、C是iid，A>B && A>C 的概率是1/3。
1/2*1/2*1/3 = 1/12
假设A>1/2，C<1/2，概率分别为1/2，此时A、B是iid，A>B 的概率是1/2。
1/2*1/2*1/2 = 1/8
1/12+1/8 = 5/24 | f*******p
发帖数: 704 | 13 P(A>B,A>C) = int_0^1 dA int_0^A dC int_(1/2)^A 2dB
= 1/6 | y***d
发帖数: 2330 | 14 你的结果是对的，只是我对条件概率这个定义不清楚，忘了怎么说的了
【在 t****a 的大作中提到】
: 注意条件概率密度D(A|A>C)=A在[0,1]间的积分仍然为1，所以应该是2A。
| q**r
发帖数: 611 | 15 其实不用条件概率， 直接积分算体积可能更直观一点。 | v********w
发帖数: 136 | 16 这个够直观
【在 h**6 的大作中提到】
: 纯直观的解：
: 假设A>1/2，C>1/2，概率分别为1/2，此时A、B、C是iid，A>B && A>C 的概率是1/3。
: 1/2*1/2*1/3 = 1/12
: 假设A>1/2，C<1/2，概率分别为1/2，此时A、B是iid，A>B 的概率是1/2。
: 1/2*1/2*1/2 = 1/8
: 1/12+1/8 = 5/24
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