Okay, so let's have this topic. So for this probability mass function or the PMF, then we have this given. So given a discrete probability distribution and to convert this to probability mass function or the PMF.
So observe that in the given distribution, we only have one probability. That's 1 over 5. So the probability is 1 over 5. If x equals 1, 2, 3, 4, and 5. So x equals 1, 2, 3, 4, and 5. Then the probability is equal to 0. If x is not equal to 1, 2, 3, 4, and 5, or the probability is equal to 0, otherwise. So, this is now the probability mass function for this distribution.
Then, for this given, so to write this in a probability mass function, or the PMF, So, observe that in the given distribution, the probability is 1 over 9 for x equals 1, 3, and 5. So, the probability is 1 over 9 if x equals 1, 3, and 5. Then the probability is equal to 3 over 9 if x equals 2 and 4. So the probability is 3 over 9 if x equals 2 and 4. Then the probability is equal to 0 if x is not equal to 1. 2, 3, 4, and 5 or 0 otherwise. So, this is now the PMF for this given distribution. Then, for this given, so, to write this to probability must function. So, observe that in the given, the probability is 1 over 7 if x equals 1 and 3. So, 1 over 7 if x equals 1 and 3, then the probability is 2 over 7. So, 2 over 7 if x equals 2 and also 3 over 7. So, 3 over 7. If x equals 0, then the probability is equal to 0 otherwise.
So, this is now the probability mass function for this distribution. Then, for this given, to show that f of x or the probability of x is a PMF. So, for this given, First, we need to substitute the values of x to this equation. So, let's start with x equals 1. So, this f of x becomes f of 1 because x is equal to 1. So, we have 1 over 8 times 1 because x is equal to 1. Then, simplify.
So, this becomes... 1 over 8. Then, for x equals 3. So, that's f of 3 equals 1 over 8 times 3. Because x is equal to 3. So, this becomes 3 over 8. Then, the last one, x equals 4. So, f of 4. And, we have 1 over 8. times 4, so we have 4 over 8. Then, to find if this given is a PMF or not, So, we need to add all these probabilities. So, we have 1 over 8 plus 3 over 8 plus 4 over 8. Then, to simplify.
So, since the same denominator, so just copy 8. Then, you add the numerators. So, 1 plus 3, that's 4. Plus 4, this one is 8. And 8 divided by 8, that's equal to 1. So, since the sum of the probabilities is equal to 1, and one properties of this PMF, that the summation of f of x is equal to 1. So, since equal to 1, so therefore, this given is... a PMF. Then, for this given, so again, we need to substitute these values of x to this equation. So, let's start with f of 0 or x is equal to 0. So, that's 1 over 35. Then, 0 squared and 0 times any number that's equal to 0. Then, for x equals 1, so f of 1, and we have 1 over 35. Then, x is 1, so 1 squared.
So, 1 squared is 1 times 1 over 35, so this is 1 over 35. Then, for x equals 3, so f of 3. And this becomes 1 over 35 times 3 to the power 2. And this 3 to the power 2, that's 9. So we have 1 over 35 times 9. And 9 times 1 over 35. So we just need to multiply these numerators and these denominators. So this becomes 9 over 35. Then, for x equals 5, so f of 5, and that's 1 over 35 times x is 5, so 5 squared. Then, to simplify, so 1 over 35 times this 5 squared is 25. Then multiply, so over 1. So multiply the numerators and these denominators. So we have 25 over 35. Then to show that this given is a PMF or not a PMF, so we need to add all these probabilities. So we have 0 plus, 1 over 35 plus 9 over 35, then plus 25 over 35. Then, since the same denominator, so just copy, we have 35. Then, you add 1 plus 9, that's 10, plus 25, that's 35. And 35 divided by 35, that's 1. So, since the sum of this f of x is equal to 1, so therefore, this given is a PMF.
Then, for this given, so to show that this given is a PMF or not, so again, substitute these values of x. So, start with f of 0. And x is equal to 0. So 0 over 15. This one is 0. Then f of 2. So this becomes 2 over 15. Then f of 4. And that's 4 over 15. Then the last one, f of 6. And that's 6 over 15. Then, add all these probabilities. So, we have 0 plus 2 over 15 plus 4 over 15. Then, plus 6 over 15. Then, copy the denominator and add the numerators.
So, 2 plus 4, that's 6, plus 6, that's 12. Since 12 divided by 15 is not equal to 1, so therefore, this given is not a PMF. Then, for this last one, So, x equals 1. So, f of 1 equals 2 times 21 times 1. So, x is equal to 1. And this becomes 2 over 21. Then, for f of 2. So, we have 2 over 21 times 2. Then, you multiply. So, over 1, multiply. So, that's 4 over... 21. So 2 times 2, that's 4. Then for 3, so f of 3. So 2 over 21, then times 3. Then you multiply.
So 2 times 3, that's 6. Then over 21. And the last one, the f of 4. So 2 over 21 times 4. So 2 times 4, that's 8, then over 21. Then add all these probabilities. So we have 2 over 21 plus 4 over 21 plus 6 over 21 plus 8 over 21. Then just copy the denominator and add the numerators. So 2 plus 4, that's 6, plus 6, that's 12, and 12 plus 8, that's 20. Since 20 over 21 is not equal to 1, so therefore this given is not a PMF.