Probability Mass Function: Overview, Questions, Preparation

The probability mass function is a method likely to correspond to any value for a discrete random variable. Moreover, this is a probability function that carries out the distribution of a discrete random variable, where the PMF is said to be positive for real numbers when the argument is non-zero and belongs to the set of random variables. Also, PMF is a function related to the probability that discrete events are correlated with certain occurring events.

Probability is a mathematical concept that explains how probable or likely an event is in occurring. The likelihood of an occurrence is a number between 0 and 1, where, in basic terms, 0 indicates the uncertainty of the event and 1 indicates the full possibility of the event.

The probability mass function is also the primary way of determining an unambiguous distribution of probabilities, and these functions arise for random variables with a scalar or multivariate domain. 

Significance of Probability Mass Functions

1. The probability mass function uses discrete values to find the probability value in Poisson and the binomial distribution.
2. PMF incorporates the random number variable, which corresponds to the random variable probability.
3. PMF plays a crucial role in the field of statistics.
4. The discrete distribution mean and its variance are calculated using the probability mass function.

The topic ‘Probability Mass Function’ is from the chapter ‘Probability’ in class 12 maths. The chapter holds 8 marks in the final exams.

1. From the below given PMF, find the value of r.

Solution. 
We know, ∑P(y_i)=1
Thus,
0 + 3r + 3r + r² = 1
6r + 6r² - 1 = 0
6r² + 6r – r -1 = 0
6r(r + 1) -1(r + 1) = 0
(6r – 1) ( r + 1 ) = 0
So, 6r – 1 = 0 and r + 1 = 0
r = -1 is not possible as the value of probability lies between 0 and 1.
Thus, r = ⅙
Therefore, the value of r from the above stated PMF is ⅙.

2.From the below given PMF, find the value of y.

Solution.
We know, ∑P(x_i)=1
Thus,
0 + 5r + 5r + 10r² = 1
10r + 10r² - 1 = 0
10r² + 10r – r -1 = 0
10r(r + 1) -1(r + 1) = 0
(10r – 1) ( r + 1 ) = 0
So, 10r – 1 = 0 and r + 1 = 0
r = -1 is not possible as the value of probability lies between 0 and 1.
Thus, r = 1/10
Therefore, the value of r from the above stated PMF is 1/10.

3. From the below given PMF, find the value of r.

Solution.
We know, ∑P(z_i)=1
Thus,
0 + 2r + 2r + 4r² = 1
4r + 4r² - 1 = 0
4r² + 4r – r -1 = 0
4r(r + 1) -1(r + 1) = 0
(4r – 1) ( r + 1 ) = 0
So, 4r – 1 = 0 and r + 1 = 0
r = -1 is not possible as the value of probability lies between 0 and 1.
Thus, r = 1/4
Therefore, the value of r from the above stated PMF is ¼.

Q: State the probability mass function of the Poisson distribution.

Q: What is the formula of Poisson distribution?

Q: What is discretisation?

Q; What is a multivariate case?

Q: What is the value of the random variable that has the largest probability mass called?