# Probability Theory 2: Finding Mean and Variance of X_n on Real Line

• Milky
So if you want to know Var(Xn), what do you need to know?The value of Xn.Okay, so now you know that you need to know Xn to know Var(Xn). But what else do you need to know?The value of b. Correct. So to summarize, in summary, the individual traveling on the real line is trying to reach the origin, with the variance of each step being \beta x^2. To find E[X_n], simply take the mean of the starting position x_0. To find Var(X_n), you will need to know the value of X_n and the value of \beta.
Milky
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!

Last edited:
Did you mean:

"whenever the person is at location xn, he next moves to a location xn+1 with error having mean 0"?

Last edited:
No, I copied exactly what the book said. I'm so confused!

Okay. Part a is easy. The problem gives away the answer. (The answer is stated unambiguously as part of the problem statement.)

For part b, what does the definition of variance say? (E.g. Var[x] = b x^2.)

Okay, so for part a, E[Xn] = 0 because with every step the mean is 0...right?

For part b.. I'm still not sure.
I know Var(X) = bx^2, but I don't know how to get from X to Xn. I tried using the definition of variance: E[X^2] - (E[X])^2 but it didn't get me very far

I know Var(X) = bx^2, but I don't know how to get from X to Xn.
The question is, if you know Xn, do you need to know X0, ..., X(n-1) to know Var(Xn)?

The variance is the same for each step that he takes, so why wouldn't the variance just still be bx^2?

No, the variance changes with each step, unless you happen to stay where you are (which is highly improbable). Look at the definition of the variance. It maps x to bx^2. Suppose Xn = t. Where does it map t?

bXn^2 ?

Yes.

## 1. What is the formula for finding the mean of X_n on the real line?

The formula for finding the mean of X_n on the real line is E[X_n] = ∑x * P(X_n = x), where x represents the possible values of X_n and P(X_n = x) represents the probability of X_n taking on that value.

## 2. How do you calculate the variance of X_n on the real line?

The variance of X_n on the real line can be calculated using the formula Var(X_n) = E[(X_n - μ)^2], where μ represents the mean of X_n. This formula can also be written as Var(X_n) = ∑(x - μ)^2 * P(X_n = x).

## 3. Can the mean and variance of X_n on the real line be negative?

Yes, the mean and variance of X_n on the real line can be negative. This is because the mean and variance are simply mathematical measures of central tendency and spread, and can take on any numerical value.

## 4. How does increasing the sample size impact the accuracy of the mean and variance of X_n on the real line?

Increasing the sample size typically leads to a more accurate estimation of the mean and variance of X_n on the real line. This is because as the sample size increases, the data used to calculate the mean and variance becomes more representative of the entire population.

## 5. What is the relationship between the mean and variance of X_n on the real line?

The mean and variance of X_n on the real line are both measures of central tendency and dispersion, but they measure different aspects of the data. The mean represents the average value of X_n, while the variance measures how spread out the data points are from the mean. In general, as the variance increases, the data points are more spread out from the mean.

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