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THE GIVEN ANSWER KEY IS 0.65 to 0.71 I AM GETTING 0.40...

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Cumulative distribution function of a random variable x means :

If a value of  a random variable is F(x) then it means probability the random variable takes any value <= x is given by F(x)..

So  to find  P(1/4  <  x  <  1)   ,  we need to find F(1) and F(1/4) and then perform F(1) - F(1/4)..

So according to the piecewise definition of function given in the question :

F(1)    =   3/4 [ Here we take 3/4 and not 1 bcoz we are asked the probability of x < 1 and not x <= 1 ]

F(1/4)  =  (1/4)2    =    1/16

Hence    P(1/4  <  x  <  1)    =    3/4  -   1/16

                                          =    11/16

                                          =    0.6875  =  0.69(correct to 2 decimal places)

Hence  0.69 should be the required probability..

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