Statistics Examples

$\begin{array}{c} x & P (x) \\ 1 & 0.4 \\ 5 & 0.1 \\ 8 & 0.2 \\ 1 & 0.1 \\ 14 & 0.2 \end{array}$

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

$0.4$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.4$ is between $0$ and $1$ inclusive

Step 3

$0.1$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.1$ is between $0$ and $1$ inclusive

Step 4

$0.2$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.2$ is between $0$ and $1$ inclusive

Step 5

$0.1$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.1$ is between $0$ and $1$ inclusive

Step 6

$0.2$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.2$ is between $0$ and $1$ inclusive

Step 7

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 8

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

$0.4 + 0.1 + 0.2 + 0.1 + 0.2$

Step 9

The sum of the probabilities for all the possible $x$ values is $0.4 + 0.1 + 0.2 + 0.1 + 0.2 = 1$ .

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Step 9.1

Add $0.4$ and $0.1$ .

$0.5 + 0.2 + 0.1 + 0.2$

Step 9.2

Add $0.5$ and $0.2$ .

$0.7 + 0.1 + 0.2$

Step 9.3

Add $0.7$ and $0.1$ .

$0.8 + 0.2$

Step 9.4

Add $0.8$ and $0.2$ .

$1$

Step 10

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: $0.4 + 0.1 + 0.2 + 0.1 + 0.2 = 1$