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Indicator function - Exercise set 1

This exercise set contains some solved exercises on indicator functions. The theory needed to solve these exercises is introduced in the lecture entitled Indicator functions.

Exercise 1.1

Consider a random variable X and another random variable Y defined as a function of X.[eq1]

Express Y using the indicator functions of the events [eq2] and [eq3].

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Denote by [eq4]the indicator of the event [eq5] and denote by [eq6]the indicator of the event [eq3]. We can write Y as:[eq8]

Exercise 1.2

Let X be a positive random variable, i.e. a random variable that can take on only positive values. Let $c$ be a constant. Prove that [eq9]where [eq10] is the indicator of the event [eq11].

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First note that the sum of the indicators [eq10] and [eq13] is always equal to 1:[eq14]As a consequence, we can write:[eq15]Now, note that [eq16] is a positive random variable and that the expected value of a positive random variable is positive:[eq17]Thus:[eq18]

Exercise 1.3

Let E be an event and denote its indicator function by $1_{E}$. Let $E^{c}$ be the complement of E and denote its indicator function by $1_{E^{c}}$. Can you express $1_{E^{c}}$ as a function of $1_{E}$?

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The sum of the two indicators is always equal to 1:[eq19]Therefore:[eq20]

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