each of ncustomers gives a hat to a hat-check person at a restaurant. the hat-check person gives the hats back to the customers in a random order. what is the expected number of customers that get back their own hat? derive your answer by using random variables and expected values of random variables.

A random variable is called with an unspecified amount or a function that gives values to each of the results of an experiment.
There are two types of random variables: discrete (having specified values) and continuous (value in a continuous range).

Utilizing indicator random variables, we resolve this issue. For 1 ≤ i ≤ n,

Let .

Xi = I {Customer 'i' who gets his hat back itself}.

The number of clients that receive their own hats back should be the random variable X. We want to figure out E[X].

Clearly,

n

X = ∑ Xi

i=1

It is clear that 1/n represents the likelihood that customer 'I' will receive his own hat.

Due to the fundamental characteristics of indicator random variables, this means that E[Xi] = 1/n.

n

E[X] = E [ ∑ Xi ]

i=1

n

= ∑ 1/n

i=1

Put the limits.

E[X] = 1

As a result, 1 consumer is anticipated to have their own hats back.

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