3.2 Probability Distributions for Discrete Random Variables

• Probabilities assigned to outcomes in $\mathcal{E}$:
  • Determine probabilities associated with values of a random variable $X$
• Probability distribution of $X$:
  • Describes how total probability of 1 is distributed among possible $X$ values

Example

• Business purchases four laser printers
  • Let $X$ = number of printers requiring service during warranty period
  • Possible $X$ values:
    • $0,1,2,3,4$
• Probability distribution details:
  • Subdivides probability of 1 among five possible values
  • Specifies probability associated with each $X$ value:
    • $p(0)$ = probability of $X=0$ = $P(X=0)$
    • $p(1)$ = probability of $X=1$ = $P(X=1)$
    • And so on
• General notation:
  • $p(x)$ = probability assigned to the value $x$

EX 3.7 number of computers in use

probability distribution

The probability distribution or probability mass function (pmf) of a discrete rv is defined for every number $x$ by

$$\begin{array}{rl}p\left(x\right)& =P\left(X=x\right)\\ & =P(\text{all }\omega \in \mathcal{S}:X\left(\omega \right)=x).\end{array}$$

• Probability mass function (pmf):

  • Specifies the probability of observing each possible value $x$ of the random variable when the experiment is performed

• Required conditions for any pmf:

  • $p(x)\ge 0$
  • ${\sum }_{\text{all possible }x}p(x)=1$

• Probability mass function (pmf) of $X$:

  • In the previous example, it was provided in the problem description

• Next steps:

  • Consider several examples
    • Exploit various probability properties
    • Obtain the desired distribution

• EX 3.8 defective components in a box

• EX 3.9 laptop or desktop

• EX 3.10 five blood donors

• Name “probability mass function”:

  • Suggested by a model used in physics for a system of “point masses”

• Model description:

  • Masses are distributed at various locations $x$ along a one-dimensional axis

• Role of the pmf:

  • Describes how the total probability mass of 1 is distributed:
    • At various points along the axis of possible values of the random variable
    • Indicates where and how much mass is located at each $x$

• Pictorial representation of a pmf:

  • Called a probability histogram
  • Similar to histograms discussed in Chapter 1

• Construction of the probability histogram:

  • Above each $y$ with $p(y)>0$:
    • Construct a rectangle centered at $y$
  • Height of each rectangle:
    • Proportional to $p(y)$
  • Base width:
    • Same for all rectangles
    • Often chosen as the distance between successive $y$ values (though it could be smaller)

• Example:

  • Figure 3.4 shows two probability histograms

Figure 3.4 Probability histograms: (a) Example 3.9; (b) Example 3.10

It is often helpful to think of a pmf as specifying a mathematical model for a discrete population.

EX 3.11 number of individuals in a household

• Population model:
  • Purpose:
    • Compute values of population characteristics
      • Example: Mean $\mu$
    • Make inferences about such characteristics
• 3.2.1 A Parameter of a Probability Distribution
• 3.2.2 The Cumulative Distribution Function
• EXERCISES Section 3.2 (11-28)