# Probability Theory 2: Finding E[X_n], Var(X_n)

• Milky
If anyone has any ideas or suggestions, I would greatly appreciate it.In summary, an individual is trying to reach the origin while traveling on the real line. The larger the desired step, the greater the variance in the result. Each step has a mean of 0 and a variance of \beta x^2. The position of the individual after n steps is denoted by X_n, with an initial position of x_0. To solve for E[X_n] and Var(X_n), properties of expectation and variance may be used. Any suggestions on how to approach this problem would be appreciated.
Milky

## Homework Statement

An individual traveling on the real line is trying to reach the origin. However, the larger the desired step, the greater is the variance in the result of that step. Specifically, whenever the person is at location x, he next moves to a location having mean 0 and variance $$\beta x^2$$. Let $$X_n$$ denote the position of the individual after having taken n steps. Supposing that $$X_0 = x_0$$, find
a. $$E[X_n]$$
b. $$Var(X_n)$$.

I am not sure how to even start this problem, and would really appreciate any suggestions!

## The Attempt at a Solution

I think I may have to use the properties of expectation and variance, but I am not sure how to even start this problem.

## 1. What is the formula for finding the expected value of a random variable, E[X_n]?

The formula for finding the expected value of a random variable is E[X_n] = ∑ x_i * P(X = x_i), where x_i represents each possible value of the random variable and P(X = x_i) represents the probability of that value occurring.

## 2. How is the variance of a random variable, Var(X_n), calculated?

The variance of a random variable is calculated using the formula Var(X_n) = E[(X_n - E[X_n])^2], where E[X_n] is the expected value of the random variable.

## 3. What is the significance of the expected value and variance in probability theory?

The expected value and variance are important measures in probability theory as they provide information about the central tendency and variability of a random variable. The expected value represents the average outcome of a random variable, while the variance measures how much the values of the random variable deviate from the expected value.

## 4. How does the concept of "law of large numbers" relate to finding E[X_n] and Var(X_n)?

The law of large numbers states that as the number of trials or observations increases, the sample mean will approach the true mean of the population. This concept is relevant to finding E[X_n] and Var(X_n) as these measures are based on probabilities and are more accurate when calculated using a larger sample size.

## 5. Can you provide an example of using probability theory to find E[X_n] and Var(X_n)?

Yes, for example, if we have a random variable X that represents the number of heads when flipping a fair coin twice, the expected value E[X] can be calculated as (0 * 0.25) + (1 * 0.5) + (2 * 0.25) = 0.5. The variance Var(X) can be calculated as ((0-0.5)^2 * 0.25) + ((1-0.5)^2 * 0.5) + ((2-0.5)^2 * 0.25) = 0.75. This means that on average, we can expect to get 0.5 heads when flipping a fair coin twice, and the values of our outcomes will vary from this expected value by approximately 0.75.

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