Chapter 5 - Probability: Random variables
Probability: Random variables
Random variable: Usually written $X$;
Is a variable (or a function whose possible values are numerical quantities) that take random vaules (from a sample set) corresponding to outcomes of a random phenomenon.
\[X \; : \; \Omega \rightarrow \mathbb{R}\] \[\omega \rightarrow X(\omega) = x\]Example 2:
- $\Omega = \{ \text{ Students of the ECE-313 class } \}$
- Experiment: “Choose randomly a student of the class and measure the height of that student”
- $X =$ The height of the student
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\[\mathbb{P}(160 \le X \le 180) = \; ?\]Probablity mass function (PMF):
The probability mass function is $\mathbb{P}_{X}(.)$ and called the probability distribution or the probability law
Example:
Flipping a die of 6 faces. If the result is even, we earn \$2, if the result is 1, we earn \$3, and if the result is 3 or 5, we lose \$4. What is the probability distribution of the random variable $X$ which gives the gain of this game.
- $\Omega = \{1, \; 2, \; 3, \; 4, \; 5, \; 6\}$
- $X = \{\$2, \; \$3, \; \$-4\}$
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