Consider a graph in Gn, p, with p=1 / n. Let X be the number of triangles in the graph, where a

step 1 So in this question we are given a random variable x. Now expectation of x is 3 and expectation

Consider a graph in Gn, p, with p=1 / n. Let X be the number...

Key Concepts

Asymptotic Analysis

Asymptotic analysis studies the behavior of functions as an argument (here, the number n of vertices) tends to infinity. In this problem, it is used to demonstrate that even as n grows large, the probability of having at least one triangle does not vanish but is bounded below by a constant (at least 1/7). This kind of analysis is essential in understanding phase transitions and limiting behaviors in random graphs.

Conditional Expectation Inequality

This inequality is a tool that allows us to relate the expectation of a random variable to its conditional expectations, often providing sharper bounds than those obtained from simple global considerations. Here it is used as a hint to derive a lower bound on the probability of finding at least one triangle in the graph, especially in the asymptotic (n ? ?) case.

Moment Method and Markov’s Inequality

The moment method involves using the expectation (and sometimes higher moments) of a random variable to derive probabilistic bounds. In particular, Markov’s inequality bounds the probability that a nonnegative random variable is at least 1 by its expectation. In this problem, since the mean number of triangles is computed to be about 1/6, Markov’s inequality directly gives an upper bound on the probability that at least one triangle is present.

Subgraph Counting

Subgraph counting is the process of counting the number of copies of a fixed subgraph (in this case, a triangle) that appear within a larger graph. This method is central when analyzing the prevalence or absence of certain patterns, and it typically involves summing indicator random variables corresponding to each possible occurrence of the subgraph.

Erd?s–Rényi Random Graph Model

This concept refers to a probability space of graphs on n labeled vertices where each of the possible edges is included independently with a given probability p. In the provided question, G(n, p) with p = 1/n is used. The model is fundamental in probabilistic combinatorics and is used to study typical properties of graphs when edges are randomly present.

Indicator Random Variables and Expectation

These are used to represent whether a certain subgraph (a triangle) appears in a given set of vertices. Each indicator is 1 if the triangle exists and 0 otherwise. The total number of triangles is the sum of these independent indicators, and by linearity of expectation, one can easily compute its mean, which plays a key role in obtaining bounds on probabilities.

Transcript

00:01 So in this question we are given a random variable x.

00:05 Now expectation of x is 3 and expectation of x square is 13.

00:11 We are asked to find what is the lower bound for the quantity given by that is probability of x greater than minus 2 and x less than 8.

00:23 So the idea here will be that we will use shebyshev's inequality and recall what shebyshev's inequality says it says that the probability of getting of realising the random variable x very far away from the mean new very far in the sense greater than k sigma will be less than 1 by k square so actually this so this is what shebi says inequality says says so intuitively it says that if your expectation is mu you are more likely to be within some distance from you so the probability of you being further away from mu is bounded by 1 by k square so we'll use the shabish as inequality so first of all we'll calculate the variance of x which is expectation of x square minus expectation of x the whole square so that will turn out to be 4 and variance of x is sigma square and if sigma square is 4 this implies that sigma is equal to 2.

01:53 Next, this is the probability which you want to find x lying between minus 2 and 8.

02:01 So this can be written as mod of x minus 3 is less than 5.

02:06 So if you expand this, this would imply x is less than 5 plus 3.

02:15 And x is greater than minus 5 plus 3.

02:20 So this will be 8 and this will be minus 2.

02:24 So this is same as this.

02:26 So this can be written as 1 minus the probability of x minus 3 being greater than 5.

02:32 So this is just the complement of this.

02:39 So we can write this.

02:45 Now probability of x minus 3 greater than 5.

02:48 Now we'll use the shabby shabing quality.

02:51 So this says that it will be less than 1 by k square...