Some answers.....
Here are my answers to some of the following questions:
1. The sum of all the digits must a multiple of 9.
4. Let En denotes the event that after n steps one returns back to the starting point for the first time. And I find P(En)=1/2 for all the n.
Proceeding in thi way: first, the number of steps one take to return to the starting point must be even, say 2, 4, 6........ We start from P(E2)=1/2. Then E4 means after the first 2 steps one must be in the diagonal point (for example, if you start from A, then this means after the first 2 steps you much arrive at C, otherwise you get back to A in 2 steps). Then from C to get back to A in 2 steps, the probability is 1/2. Thus P(E4)=1/2. You can also reason like this, you have two different paths to reach C from A in 2 steps, and for each of this path, you have two paths for another 2 steps, one of each leading you back to A, and one leading you to C. So there are 4 paths in all for 4 steps, and 2 of which gets back to A, two of which gets to C (you cannot arrive at B and D in an even number of steps). You can proceed similarily for other n. So the expectation does not converge.
5. If p is a prime number larger than 3, then P^2-1 should be a multiple of 24.
First p must be odd, so let p=2n-1 or 2n+1, then p^2-1=(p-1)(p+1)=(2n-2)2n or 2n(2n+2), so 4(n-1)n or 4n(n+1), the product of two consecutive numbers is even, so this means p^2-1 must a multiple of 8.
Secondly, p must not a muliple of 3, so p=3n+1 or 3n+2, then p^2-1= 3n(3n+2) or (3n+1)(3n+3), in either case, p^2-1 is a multiple of 3. So it is a multiple of 24.
6. f(n)=f(n-1)+f(n-2), where n is the total number of steps, f(n) is the number of possible ways, and the boundary condition is f(1)=1, f(2)=2.
We can focus on the last jump, it can either be 1 step or 2 steps. If it is a 1 step jump , then the number of ways is determined by the number of ways for the first n-1 steps, that is f(n-1). Similarily, if it is a 2 step jump, then the number of ways is determined by the first n-2 steps, that is f(n-2). No other choice for the last step jump, so the totol number is just the sum, that is, f(n)=f(n-1)+f(n-2).
7. Simple statistics, var(Z)=a^2Var(X)+b^2Var(Y)+2abVar(X)Var(Y)Rho, wher Rho is the correlation coefficient of X and Y.
Here are my answers to some of the following questions:
1. The sum of all the digits must a multiple of 9.
4. Let En denotes the event that after n steps one returns back to the starting point for the first time. And I find P(En)=1/2 for all the n.
Proceeding in thi way: first, the number of steps one take to return to the starting point must be even, say 2, 4, 6........ We start from P(E2)=1/2. Then E4 means after the first 2 steps one must be in the diagonal point (for example, if you start from A, then this means after the first 2 steps you much arrive at C, otherwise you get back to A in 2 steps). Then from C to get back to A in 2 steps, the probability is 1/2. Thus P(E4)=1/2. You can also reason like this, you have two different paths to reach C from A in 2 steps, and for each of this path, you have two paths for another 2 steps, one of each leading you back to A, and one leading you to C. So there are 4 paths in all for 4 steps, and 2 of which gets back to A, two of which gets to C (you cannot arrive at B and D in an even number of steps). You can proceed similarily for other n. So the expectation does not converge.
5. If p is a prime number larger than 3, then P^2-1 should be a multiple of 24.
First p must be odd, so let p=2n-1 or 2n+1, then p^2-1=(p-1)(p+1)=(2n-2)2n or 2n(2n+2), so 4(n-1)n or 4n(n+1), the product of two consecutive numbers is even, so this means p^2-1 must a multiple of 8.
Secondly, p must not a muliple of 3, so p=3n+1 or 3n+2, then p^2-1= 3n(3n+2) or (3n+1)(3n+3), in either case, p^2-1 is a multiple of 3. So it is a multiple of 24.
6. f(n)=f(n-1)+f(n-2), where n is the total number of steps, f(n) is the number of possible ways, and the boundary condition is f(1)=1, f(2)=2.
We can focus on the last jump, it can either be 1 step or 2 steps. If it is a 1 step jump , then the number of ways is determined by the number of ways for the first n-1 steps, that is f(n-1). Similarily, if it is a 2 step jump, then the number of ways is determined by the first n-2 steps, that is f(n-2). No other choice for the last step jump, so the totol number is just the sum, that is, f(n)=f(n-1)+f(n-2).
7. Simple statistics, var(Z)=a^2Var(X)+b^2Var(Y)+2abVar(X)Var(Y)Rho, wher Rho is the correlation coefficient of X and Y.
