A random variable X takes values -1,0,1,2 with probabilities 1+3 p 4 , 1-p 4 , 1+2 p 4 , 1-4 p 4 respectively, where p varies over R . Then the minimum and maximum values of the mean of X are respectively.

Step-by-step solution

Here , 1+3 p 4 , 1-p 4 , 1+2 p 4 and 1-4 p 4 are probabilities when X takes values -1,0,1 and 2 respectively. Therefore, each is greater than or equal to 0 and less than or equal to 1 . aligned & i.e., 0 1+3 p 4 1,0 1-p 4 1 & 0 1+2 p 4 1 and 0 1-4 p 4 1 & - 1 3 p 1 4 aligned aligned Mean (X)=-1 1+3 p 4 +0 1-p 4 & +1 1+2 p 4 & +2 1-4 p 4 aligned = 2-9 p 4 Now, - 1 3 p 1 4 aligned & 3 -9 p - 9 4 & - 1 4 2-9 p 5 & - 1 16 2-9 p 4 5 4 aligned