# Thread: NO idea on how to do these two questionsi

1. ## NO idea on how to do these two questionsi

1. Consider the set V = {1,2,...,n} and let p be a real number with 0 < p < 1. We construct a graph G = (V, E) with vertex set V , whose edge set E is determined by the following random process: Each unordered pair {o,j} of vertices, where i can't j, occurs as anedge in E with probability p, independently of the other unordered pairs. A triangle in G is an unordered triple {i,j,k} of distinct vertices, such that {i,j} , {j,k}, and {k,i} are edges in G.

De ne the random variable X to be the total number of triangles in the graph G. Determine the expected value E(X).

Note: I've tried a bunch of things and just get stuck I'm pretty sure You use indicator random variables. Just I'm having a mind fart.

2. ## Re: NO idea on how to do these two questionsi

Originally Posted by oreo
1. Consider the set V = {1,2,...,n} and let p be a real number with 0 < p < 1. We construct a graph G = (V, E) with vertex set V , whose edge set E is determined by the following random process: Each unordered pair {o,j} of vertices, where i can't j, occurs as anedge in E with probability p, independently of the other unordered pairs. A triangle in G is an unordered triple {i,j,k} of distinct vertices, such that {i,j} , {j,k}, and {k,i} are edges in G. Dene the random variable X to be the total number of triangles in the graph G. Determine the expected value E(X).
I admit that I do not fully understand the setup here.

But there are some ways to simply the problem.
One is to think of selecting n points on a unit circle as the set $V$. This avoids three colinear points.

Let $N=\dbinom{n}{2}$. Then there are $2^N$ possible graphs on those points.

Let $M=\dbinom{n}{3}$. Then there are $2^M-1$ possible triangles on those points.

Now any for any of those triangles to exist the probability is $p^3$ because each edge must be present.

Having confessed that I have not thought this through completely, I hope this gives you some ideas.

3. ## Re: NO idea on how to do these two questionsi

ya this one is really annoying hahaha not sure what to do with it exactly