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GO Classes DPP | STATISTICS | VARIANCE

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Amandeep is in the middle of a bridge of infinite length. He takes the unit step to the right with probability $p$ and to the left with probability $1-p$. Assume that the movements are independent of each other.
Hint: Consider the random variable $X_{i}$ associated with the $i^{\text {th }}$ step defined as:
$$
X_{i}= \begin{cases}1 & \text { if the step of Amandeep is towards the right } \\ -1 & \text { if the step of Amandeep is towards the left }\end{cases}
$$

What is the variance distance between the starting point and end point of Amandeep after $n$ steps?
A. $4 n p(1-p)$
B. $4 p(1-p)$
C. 1
D. $n^{2}(2 p-1)^{2}-1$
 

1 Answer

1 1 vote
Let us associate a random variable $X_{i}$ with the $i^{\text {th }}$ step.
$$
\begin{aligned}
& X_{i}= \begin{cases}1 & \text { if } i^{\text {th }} \text { step of Amandeep is towards the right } \\
-1 & \text { if } i^{\text {th }} \text { step of Amandeep is towards the left }\end{cases} \\
& E\left(X_{i}\right)=1 \times P\left(X_{i}=1\right)+(-1) \times P\left(X_{i}=-1\right) \\
& E\left(X_{i}\right)=1 \times p-1 \times(1-p)=2 p-1
\end{aligned}
$$
$E\left(X_{i}\right)^{2}=1^{2} \times P\left(X_{i}=1\right)+(-1)^{2} \times P\left(X_{i}=-1\right)$
$E\left(X_{i}\right)^{2}=p+(1-p)=1$
$V\left(X_{i}\right)=E\left(X_{i}\right)^{2}-\left[E\left(X_{i}\right)\right]^{2}$
$V\left(X_{i}\right)=1-(2 p-1)^{2}=4 p(1-p)$
Therefore, $V(S)=\sum_{i=1}^{n} V\left(X_{i}\right)$ (because, movements of steps are independent) Hence, $V(S)=\sum_{i=1}^{n} 4 p(1-p)=4 n p(1-p)$
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