 Good morning friends. I am Poorva and today we will work out the following question. State which of the following are not the probability distributions of a random variable. Give reasons for your answer. Now in a probability distribution, some of all probabilities is 1. So this is the key idea behind our question. Let us begin with the solution now. Now this is the probability distribution table in the first part. Now in this table we are given that when x is equal to 0 probability of x is equal to 0.4. When x is equal to 1 probability of x is equal to 0.4. And when x is equal to 2 probability of x is equal to 0.2. Now by key idea we know that in a probability distribution some of all probabilities is 1. So here we will find the sum of probabilities. Now the sum of probabilities that is px is equal to 0.4 plus 0.4 plus 0.2. And this comes out to be equal to 1. Now since the sum of probabilities is equal to 1. So we have thus this is a probability distribution of a random variable. Now this is the table given to us in the second part. Here we have when x is equal to 0 probability of x is equal to 0.1. When x is equal to 1 probability of x is equal to 0.5. When x is equal to 2 probability of x is equal to 0.2. When x is equal to 3 probability of x is equal to minus 0.1. And when x is equal to 4 probability of x is equal to 0.3. Now sum of the probabilities that is px is equal to 0.1 plus 0.5 plus 0.2 plus of minus 0.1 plus 0.3. And we get this is equal to 1. Now here we have when x is equal to 3 probability of x is equal to minus 0.1. But probability cannot be negative. Therefore this is not a probability distribution of a random variable. This is our third part. Here we have when y is equal to minus 1 probability of y is equal to 0.6. When y is equal to 0 probability of y is equal to 0.1. And when y is equal to 1 probability of y is equal to 0.2. Now sum of the probabilities that is probability of y is equal to 0.6 plus 0.1 plus 0.2 And we get this is equal to 0.9. Now by key idea we know that in a probability distribution sum of all probabilities is 1. So here we have since the sum of the probabilities is equal to 0.9. So this is not a probability distribution of a random variable. Now this is our fourth part. Here we have when z is equal to 3 probability of z is equal to 0.3. When z is 2 probability of z is 0.2. When z is 1 probability of z is 0.4. When z is 0 probability of z is 0.1. And when z is equal to minus 1 probability of z is equal to 0.05. Now the sum of the probabilities that is probability of z is equal to 0.3 plus 0.2 plus 0.4 plus 0.1 plus 0.05. And this is equal to 1.05. So again by key idea since the sum of the probabilities is not equal to 1. So we have this is not a probability distribution of a random variable. So we have got our answer as second, third and fourth. Hope you have understood the solution. Bye and take care.