The probability density function of the random variable X is defined by f ( x ) = { 4 ( x − x 3 ) , 0 ≤ x ≤ 1 0 , elsewhere What is the probability that three independent observations from the distribution of X are all less than the mode of X?This is a question I got incorrect. I have very little experience with mode and I understand that it's the value of the random variable with the highest probability. The solution for finding the mode is as follows:The max point occurs when f ′ ( x ) = 0 f ′ ( x ) = 4 − 12 x 2 = 0 x = 1 3 This is the max point or mode because f ″ ( x ) is a negative number.

Question

The probability density function of the random variable   X is defined by  f  (  x  )  =      {                            4          (          x          −                      x            3                    )          ,                          0          ≤          x          ≤          1                                      0          ,                          elsewhere                        What is the probability that three independent observations from the distribution of   X are all less than the mode of   X?This is a question I got incorrect. I have very little experience with mode and I understand that it&#039;s the value of the random variable with the highest probability. The solution for finding the mode is as follows:The max point occurs when       f    ′    (  x  )  =  0      f    ′    (  x  )  =  4  −  12      x    2    =  0  x  =            1      3      This is the max point or mode because       f    ″    (  x  ) is a negative number.

Lea Johnson · Accepted Answer

You&#039;re correct that the first derivative being 0 means that you have an extremum (maximum or minimum). The solution then mentions that f&quot;(x) is a negative number, which makes it a maximum.Remember that f&quot;(x) describes the curvature of the function. If it is positive, the function is curving &quot;up, like a cup&quot; and if it is negative, the function is curving &quot;down, like a frown.&quot; You should be able to visualize which of these would result in a local maximum.How, then, do you know it is a global maximum? The nice thing about continuous functions on closed intervals (here the interval is [0,1]) is that they will achieve their maximum value EITHER at the endpoints of the intervals, or at a local extremum, but nowhere else. So to be absolutely certain, you could check the value of the function at the endpoints as well and compare.

The probability density function of the random variable X is defined by f ( x )

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