If a random variable X has the following probability distribution of X ( array |l|c|c|c|c|c|c|c|c| X =x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 P ( X =x) & 0 & k & 2 k & 2 k & 3 k & k ^2 & 2 k ^2 & 7 k ^2+ k array ) Then P ( X 6)=

Step-by-step solution

The sum of probabilities in the distribution must be 1: P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) = 1 . Substituting the given expressions: 0 + k + 2k + 2k + 3k + k^2 + 2k^2 + (7k^2 + k) = 1 . Combining like terms yields 10k^2 + 9k = 1 , which rearranges to the quadratic equation 10k^2 + 9k - 1 = 0 . Factoring gives (10k - 1)(k + 1) = 0 , with solutions k = 1 10 or k = -1 . Since probability must be non-negative, k = 1 10 . To find P(X 6) , sum P(X=6) + P(X=7) : P(X=6) = 2k^2 = 2( 1 10 )