 Hi friends, I am Purva and today we will work out the following question. The random variable x has a probability distribution, probability of x of the following form, where k is some number. Probability of x is equal to k if x is equal to 0. Probability of x is equal to 2k if x is equal to 1. Probability of x is equal to 3k if x is equal to 2 and probability of x is equal to 0 otherwise. A we have to determine the value of k and B we have to find probability of x less than 2, probability of x less than equal to 2 and probability of x greater than equal to 2. Now in a probability distribution, sum of all probabilities is equal to 1. So this is the key idea behind our question. Let us begin with the solution now. Now based on the given data, we can write the probability distribution of x as, so this is the probability distribution of x. Now here we can clearly see that when x is 0 probability of x is k, when x is equal to 1 probability of x is equal to 2k. When x is 2 probability of x is 3k and when x has any other value then probability of x is 0. Now in the A part we have to determine the value of k. By key idea we know that in a probability distribution sum of all probabilities is 1. So here we have since the sum of all probabilities in a distribution should be 1. So we have k plus 2k plus 3k plus 0 is equal to 1. As the sum of all probabilities of x is equal to 1 or we have 6k is equal to 1 or k is equal to 1 upon 6. So we have got the value of k as 1 upon 6. Now in the B part we have to find probability of x less than 2, probability of x less than equal to 2 and probability of x greater than equal to 2. So first we will find probability of x less than 2. Now probability of x less than 2 is equal to probability of x when x is equal to 0 plus probability of x when x is equal to 1. So for probability of x less than 2 probabilities are probability of x is equal to 0 plus probability of x is equal to 1. Now probability of x is equal to 0 is k and probability of x is equal to 1 is 2k. So we get here 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 6 so we have 1 upon 6 and we get this is equal to 1 upon 2. Now we will find probability of x less than equal to 2. Probability of x less than equal to 2 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 equal to 2 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 and this is equal to k plus 2k plus 3k which is equal to 6k and this is equal to 6 into k is equal to 1 upon 6 so we have 1 upon 6 and this is equal to 1. Now finally we will find probability of x greater than equal to 2 and probability of x greater than equal to 2 is equal to probability of x when x is equal to 2 plus probability of x when x is equal to any other value. So for probability of x greater than equal to 2 probabilities are probability of x is equal to 2 plus probability of x is equal to any other value. This is equal to now probability of x is equal to 2 is 3k plus probability of x is equal to any other value is 0 and we get this is equal to 3k. This is equal to 3 into k and k is equal to 1 upon 6 and we get this is equal to 1 upon 2. So we have got answer for the a part as k is equal to 1 upon 6. Answer for the b part is probability of x less than 2 is equal to 1 upon 2. Probability of x less than equal to 2 is equal to 1 and probability of x greater than equal to 2 is equal to 1 upon 2. Hope you have understood the solution. Bye and take care.