Let X be a random variable, and let P(X=x) denote the probability that X takes the value x. Suppose that the points (x, P(X=x)), x=0,1,2,3,4, lie on a fixed straight line in the x y-plane, and P(X=x)=0 for all x R -\ 0,1,2,3,4\ . If the mean of X is 5 2 , and the variance of X is , then the value of 24 is ___
A. A
Let equation of line is y = mx + c array |c|c|c|c|c|c|c| x & 0 & l & 2 & 3 & 4 & R -\ 0,1,2,3,4\ \\ P ( x ) & c & m + c & 2 ~m + c & 3 ~m + c & 4 ~m + c & 0 \\ array _ x=0 ^4(m x+c)=1 10 m+5 c=1 2 m+c= 1 5 ...(1) aligned & mean = x_i P_i= _ i =0 ^4 ( mx _ i + c ) x _ i =30 ~m +10 c = 5 2 \\ & 3 ~m + c = 1 4 (2) aligned aligned & P _ i x _ i ^2= _ i =0 ^4 ( mx _ i + c ) x _1^2 \\ & = _ i =0 ^4 ( mx _ i ^3+ cx _ i ^2 ) 100 ~m +30 c (Now putting m and c ) \\ & P _ i x _ i ^2=5+3=8 \\ & Variance = P _ i x _ i ^2- ( P _ i x _ i )^2=8- ( 5 2 )^2= 7 4 \\ & 24 =42 aligned
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