Find the mode, we need to find the value of x for which f &#x2032; </msup>

crossoverman9b

crossoverman9b

Answered question

2022-06-10

Find the mode, we need to find the value of x for which f ( x ) = 0 and f ( x ) < 0. After applying derivative to the pdf, how do we proceed?

Answer & Explanation

Donavan Mack

Donavan Mack

Beginner2022-06-11Added 24 answers

The mode is the x such that the pdf attains its global maximum. In general, it is true that "if f ( x ) = 0 and f ( x ) < 0, then x is the global maximum of a function".
Let's turn to math.
f ( x ) = λ e λ x , for   x [ 0 , + )
f ( x ) = λ 2 e λ x
Then:
f ( x ) = 0 λ 2 e λ x = 0 no  x  satisfies   f ( x ) = 0.
Let's look at x = 0:
f ( 0 ) = λ e λ 0 = λ > 0.
Moreover, notice that:
f ( x ) > 0   x [ 0 , + ) ,
and
f ( x ) < 0   x [ 0 , + ) .
This means that this function is always positive and always decreases. Hence, you can conclude that the global maximum is attained at x = 0. For these reasons, x = 0 is the mode.

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