 Hi friends, I am poor one today. We will work out the following question a random variable x has the following probability distribution Determined first k second probability of x less than 3 third probability of x greater than 6 and 4th probability of x when x is between 0 and 3 now in a probability distribution sum of all probabilities is 1 So this is the key idea behind our question Let us begin with the solution now Now in the question we are given the probability distribution of a random variable x now from this table We can clearly see that when x is equal to 0 probability of x is 0 when x is 1 probability of x is k When x is 2 probability of x is 2k when x is 3 probability of x is 2k when x is 4 probability of x is 3k When x is 5 probability of x is k square when x is 6 probability of x is 2k square and When x is 7 probability of x is 7 k square plus k Now in the first part we have to determine this k Now by key idea we know that in a probability distribution sum of all probabilities is 1 So here in the first part we have the sum of All probabilities in a distribution should be 1 so we have 0 plus k plus 2k plus 2k plus 3k plus k square Plus 2k square plus 7k square Plus k is equal to 1 That is the sum of all these probabilities of x is equal to 1 or we have 9k plus 10k square is equal to 1 Or we can write this as 10k square Plus 9k minus 1 is equal to 0 Solving this quadratic equation we get 10k minus 1 Into k plus 1 is equal to 0 Now since probability cannot be negative therefore, we get 10k minus 1 is equal to 0 because we ignore this k plus 1 is equal to 0 And this implies k is equal to 1 upon 10 So we have got the value of k as 1 upon 10 Now in the second part we have to find probability of x less than 3 Now probability of x less than 3 is equal to probability of x when x is equal to 0 Plus probability of x when x is equal to 1 plus probability of x when x is equal to 2 So here we have for probability of x less than 3 Probabilities are Probability of x is equal to 0 plus probability of x is equal to 1 plus probability of x is equal to 2 Now probability of x is equal to 0 is 0 probability of x is equal to 1 is k and Probability of x is equal to 2 is 2k so we get This is equal to 0 plus k plus 2k Which is equal to 0 plus now k is equal to 1 upon 10 so we have 1 upon 10 plus 2 into 1 upon 10 and we get this is equal to 3 upon 10 Now in the third part we have to find probability of x greater than 6 Now for probability of x greater than 6 We have the probabilities are Probability of x is equal to 7 And from the table we can clearly see that probability of x is equal to 7 is 7k square plus k So here we get this is equal to 7k square plus k Which is equal to 7 into now k is equal to 1 upon 10 so we have 1 upon 10 whole square plus 1 upon 10 and We get this is equal to 7 into 1 upon 100 plus 1 upon 10 and This is equal to 17 upon 100 Now in the fourth part we have to determine probability of x when x is between 0 and 3 now probability of x when x is between 0 and 3 is equal to Probability of x when x is equal to 1 plus probability of x when x is equal to 2 so For probability of x when x is between 0 and 3 probabilities are Probability of x is equal to 1 plus probability of x is equal to 2 This is equal to k plus 2k Which is equal to 3k and This is equal to 3 into now k is equal to 1 upon 10 and this is further equal to 3 upon 10 So the answer for the first part is k is equal to 1 upon 10 Answer for the second part is probability of x less than 3 is equal to 3 upon 10 Answer for the third part is probability of x greater than 6 is equal to 17 upon 100 and Answer for the fourth part is probability of x between 0 and 3 is 3 upon 10. Hope you have understood the solution. Bye and take care