Finite Math Examples

$\begin{array}{c} x & P (x) \\ 0 & 0.23 \\ 1 & 0.38 \\ 2 & 0.22 \\ 3 & 0.13 \\ 4 & 0.03 \\ 5 & 0.01 \\ 6 & 0 \end{array}$

Step 1

A discrete random variable $x$ takes a set of separate values (such as $0$ , $1$ , $2$ ...). Its probability distribution assigns a probability $P (x)$ to each possible value $x$ . For each $x$ , the probability $P (x)$ falls between $0$ and $1$ inclusive and the sum of the probabilities for all the possible $x$ values equals to $1$ .

1. For each $x$ , $0 \leq P (x) \leq 1$ .

2. $P (x_{0}) + P (x_{1}) + P (x_{2}) + \dots + P (x_{n}) = 1$ .

Step 2

$0.23$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.23$ is between $0$ and $1$ inclusive

Step 3

$0.38$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.38$ is between $0$ and $1$ inclusive

Step 4

$0.22$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.22$ is between $0$ and $1$ inclusive

Step 5

$0.13$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.13$ is between $0$ and $1$ inclusive

Step 6

$0.03$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.03$ is between $0$ and $1$ inclusive

Step 7

$0.01$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.01$ is between $0$ and $1$ inclusive

Step 8

$0$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0$ is between $0$ and $1$ inclusive

Step 9

For each $x$ , the probability $P (x)$ falls between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0 \leq P (x) \leq 1$ for all x values

Step 10

Find the sum of the probabilities for all the possible $x$ values.

$0.23 + 0.38 + 0.22 + 0.13 + 0.03 + 0.01 + 0$

Step 11

The sum of the probabilities for all the possible $x$ values is $0.23 + 0.38 + 0.22 + 0.13 + 0.03 + 0.01 + 0 = 1$ .

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Step 11.1

Add $0.23$ and $0.38$ .

$0.61 + 0.22 + 0.13 + 0.03 + 0.01 + 0$

Step 11.2

Add $0.61$ and $0.22$ .

$0.83 + 0.13 + 0.03 + 0.01 + 0$

Step 11.3

Add $0.83$ and $0.13$ .

$0.96 + 0.03 + 0.01 + 0$

Step 11.4

Add $0.96$ and $0.03$ .

$0.99 + 0.01 + 0$

Step 11.5

Add $0.99$ and $0.01$ .

$1 + 0$

Step 11.6

Add $1$ and $0$ .

$1$

Step 12

For each $x$ , the probability of $P (x)$ falls between $0$ and $1$ inclusive. In addition, the sum of the probabilities for all the possible $x$ equals $1$ , which means that the table satisfies the two properties of a probability distribution.

The table satisfies the two properties of a probability distribution:

Property 1: $0 \leq P (x) \leq 1$ for all $x$ values

Property 2: $0.23 + 0.38 + 0.22 + 0.13 + 0.03 + 0.01 + 0 = 1$