Register

GO Classes DPP | STATISTICS | EXPECTATION

229 views
1 1 vote
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 expected distance between the starting point and end point of Amandeep after $n$ steps?
A. $2 p-1$
B. $n(2 p-1)$
C. $1-2 p$
D. $n(1-2 p)$

1 Answer

1 1 vote
Let us associate a random variable $X_{i}$ with the $i^{\text {th }}$ step.
$$
X_{i}= \begin{cases}1 & \text { if } i^{\text {th }} \text { step of Amandeep is towards the right } \\ -1 & \text { if } i^{t h} \text { step of Amandeep is towards the left }\end{cases}
$$
$E\left(X_{i}\right)=1 \times P\left(X_{i}=1\right)+(-1) \times P\left(X_{i}=-1\right)$
$E\left(X_{i}\right)=1 \times p-1 \times(1-p)=2 p-1$
$S=X_{1}+X_{2}+\ldots+X_{n}$ represents the random distance moved from the starting point after $n$ steps.
Therefore, $E(S)=\sum_{i=1}^{n} E\left(X_{i}\right)=n(2 p-1)$
Hence, expected distance between the starting point and end point of Amandeep after $n$ steps is $n(2 p-1)$
Voting & Accepting an answer helps improve the community
Answer:
← Previous · GO Classes DPP | STA...Next → · GO Classes DPP | STA...
← Previous in category · GO Classes DPP | STA...Next in category → · GO Classes DPP | STA...
Position:
Show:

Related questions

2 2 votes
1 1 answer
216
216 views
GO Classes asked Oct 16, 2025
216 views
GO Classes DPP | STATISTICS | EXPECTATION
An unbiased die is thrown $n+2$ times. After each throw a ' + ' is recorded for 2 or 5 and '-' is recorded for $1,3,4$ or 6 , the signs forming an ordered sequence. To ea...
1 1 vote
1 1 answer
258
258 views
GO Classes asked Oct 16, 2025
258 views
GO Classes DPP | STATISTICS | EXPECTATION
There are $2^{10}$ numbered cards in a deck among which ${ }^{10} C_{i}$ cards bear the number i ; $\mathrm{i}=0,1,2, \ldots, 10$. From the deck, 4 cards are drawn with r...
3 3 votes
1 1 answer
262
262 views
GO Classes asked Oct 16, 2025
262 views
GO Classes DPP | STATISTICS | VARIANCE
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...
1 1 vote
1 1 answer
256
256 views
GO Classes asked Oct 16, 2025
256 views
GO Classes DPP | STATISTICS | VARIANCE
An unbiased die is thrown $n+2$ times. After each throw a ' + ' is recorded for 2 or 5 and '-' is recorded for $1,3,4$ or 6 , the signs forming an ordered sequence. To ea...