Discrete Random Variable and Probability Mass Function
Data Science and A.I. Lecture Series
- A random variable is said to be discrete if it has either a finite or a countable number of values.
- Countable values are those which can be arranged in a sequence, corresponding to natural numbers.
- Example: Number of students present each day in a class.
Probability Mass Function (PMF)
Let \(X\) be a discrete random variable taking values \(x_1, x_2, …\).
The probability mass function (PMF) is defined as:
\[ P(X = x_i) = p(x_i) \]
It satisfies the conditions:
- \( p(x_i) \geq 0 \) for all \(i\).
- \( \sum_{i} p(x_i) = 1 \).
Example 1: Valid Probability Distribution?
Is the following a probability distribution?
| X | P(X) |
|---|---|
| 0 | \( \frac{1}{2} \) |
| 1 | \( \frac{3}{4} \) |
Sum: \( \frac{1}{2} + \frac{3}{4} = \frac{5}{4} > 1 \), so it is not a valid probability distribution.
Example 2: Probability Distribution
Find the constant \(c\) in the given probability distribution:
| X | P(X) |
|---|---|
| 0 | \( c \) |
| 1 | \( c \) |
| 2 | \( 2c \) |
| 3 | \( 3c \) |
| 4 | \( c \) |
Equation: \( c + c + 2c + 3c + c = 1 \).
Solving: \( 8c = 1 \Rightarrow c = \frac{1}{8} \).
Example 3: Number of Heads in 3 Coin Tosses
Find the probability distribution of the number of heads when tossing 3 fair coins.
| X | P(X) |
|---|---|
| 0 | \( \frac{1}{8} \) |
| 1 | \( \frac{3}{8} \) |
| 2 | \( \frac{3}{8} \) |
| 3 | \( \frac{1}{8} \) |
Conclusion
- A discrete random variable has countable values.
- PMF must satisfy \( p(x_i) \geq 0 \) and \( \sum p(x_i) = 1 \).
- Examples illustrated the application of PMFs.
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