
*1 point*

Amandeep is in the middle of a bridge of infinite length. He takes the unit step to the right with probability 0.66 and to the left with probability 0.34. Assume that the movements are independent of each other.
**Hint:** Consider the random variable Xi*Xi*​ associated with the ith*ith* step defined as:
Xi*Xi*​={1if the step of Amandeep is towards the right −1if the step of Amandeep is towards the left {1−1​if the step of Amandeep is towards the right if the step of Amandeep is towards the left ​
What is the expected distance between the starting point and end point of Amandeep after 5 steps?

Let's analyze the problem step by step.

### Step 1: Understanding the Random Variable \( X_i \)
Given that Amandeep is standing in the middle of a bridge of infinite length:
- He takes a unit step to the right with probability 0.66.
- He takes a unit step to the left with probability 0.34.

Let \( X_i \) be the random variable associated with the \( i \)-th step:
\[
X_i = 
\begin{cases} 
1 & \text{if the step is towards the right}, \\
-1 & \text{if the step is towards the left}.
\end{cases}
\]
So, the expected value of \( X_i \) is:
\[
\mathbb{E}[X_i] = 1 \times 0.66 + (-1) \times 0.34 = 0.66 - 0.34 = 0.32.
\]

### Step 2: Distance Calculation after 5 Steps
After 5 steps, Amandeep's position relative to the starting point can be expressed as the sum of the steps taken:
\[
S_5 = X_1 + X_2 + X_3 + X_4 + X_5.
\]
The expected distance from the starting point after 5 steps is the expected value of \( S_5 \):
\[
\mathbb{E}[S_5] = \mathbb{E}[X_1 + X_2 + X_3 + X_4 + X_5] = \mathbb{E}[X_1] + \mathbb{E}[X_2] + \mathbb{E}[X_3] + \mathbb{E}[X_4] + \mathbb{E}[X_5].
\]
Since each \( X_i \) has the same expected value:
\[
\mathbb{E}[S_5] = 5 \times \mathbb{E}[X_i] = 5 \times 0.32 = 1.6.
\]

### Step 3: Variance and Standard Deviation
The variance of each step \( X_i \) can be calculated as:
\[
\text{Var}(X_i) = \mathbb{E}[X_i^2] - (\mathbb{E}[X_i])^2.
\]
Since \( X_i \) takes values 1 or -1:
\[
\mathbb{E}[X_i^2] = 1^2 \times 0.66 + (-1)^2 \times 0.34 = 0.66 + 0.34 = 1.
\]
Thus:
\[
\text{Var}(X_i) = 1 - (0.32)^2 = 1 - 0.1024 = 0.8976.
\]
The variance of the sum \( S_5 \) is:
\[
\text{Var}(S_5) = 5 \times \text{Var}(X_i) = 5 \times 0.8976 = 4.488.
\]
The standard deviation is:
\[
\sigma_{S_5} = \sqrt{4.488} \approx 2.12.
\]

### Final Step: Expected Distance (Magnitude)
The expected distance (magnitude) from the starting point after 5 steps is the expected value of \( S_5 \), which is \( 1.6 \).

Thus, the expected distance between the starting point and the endpoint of Amandeep after 5 steps is \( \boxed{1.6} \).

Explore how to calculate the expected distance between the starting point and endpoint after 5 steps using probability theory and random variables. This problem involves determining the expected value, variance, and standard deviation of Amandeep's position on an infinite bridge. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation