Welcome to the article Expectation and Linearity 1 1 1 3 marks Linearity of expectation is often a powerful tool when other combinatorial pure counting based approaches prove cumbersome. On this page, you will learn the essential and logical steps to better understand the topic being discussed. We hope the information provided helps you gain valuable insights and is easy to follow. Let’s begin the discussion!
Answer :
the expected value of Xk is 1/n, and the expected value of X, the number of students who got back their own pens, is 1. Linearity of expectation allows us to find the expected value of X by taking the sum of the expected values of the indicator random variables Xk.
In this problem, we are asked to find the expected number of students who got back their own pens. To do this, we can use the concept of linearity of expectation.
First, let's define some terms. Let Ak be the event that the kth student got back his/her pen, and let Xk be its indicator random variable. The indicator random variable Xk takes on the value of 1 if the kth student got back his/her pen, and 0 otherwise.
Now, we want to find the expected value of Xk. The expected value of Xk is the probability that the event Ak occurs. In this case, since the pens are randomly and uniformly shuffled, the probability that the kth student gets back his/her own pen is 1/n, where n is the total number of students. Therefore, the expected value of Xk is 1/n.
Next, let's define X as the number of students who got back their own pens. We can express X as the sum of the indicator random variables X1, X2, ..., Xn. In other words, X = X1 + X2 + ... + Xn.
Using linearity of expectation, we can find the expected value of X by taking the sum of the expected values of X1, X2, ..., Xn. Since the expected value of each Xk is 1/n, we have:
E[X] = E[X1 + X2 + ... + Xn] = E[X1] + E[X2] + ... + E[Xn] = 1/n + 1/n + ... + 1/n = n * (1/n) = 1.
Therefore, the expected number of students who got back their own pens is 1.
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