# Probability Distribution Table

How to find the expected value, variance, and standard deviation from the probability distribution table: formula, 4 examples, and their solutions.

## Definition

### Definition

This is a probability distribution table.

X: Value for each case

P(X): Probability for each case

## Formula

### Formula

These are the formulas

to find E(X), V(X), and σ(X).

p_{1} + p_{2} + p_{3} = 1

The sum of the probabilities is 1.

E(X) = x_{1}p_{1} + x_{2}p_{2} + x_{3}p_{3} + ...

Expected Value

V(X) = E(X^{2}) - {E(X)}^{2}

New formula to find the variance V(X)

σ(X) = √V(X)

Standard Deviation

## Example 1

### Example

### Solution

The sum of the probabilities is 1.

So

1/4 + a + 1/3 + 1/6 = 1.

Multiply 12 to both sides.

[1/4]⋅12 = 3

+a⋅12 = +12a

+[1/3]⋅12 = +4

+[1/6]⋅12 = +2

1⋅12 = 12

3 + 4 + 2

= 7 + 2

= 9

Move 9 to the right side.

Then 12a = 3.

Divide both sides by 3.

Then 4a = 1.

Divide both sides by 4.

Then a = 1/4.

So a = 1/4.

## Example 2

### Example

### Solution

You just found that

a = 1/4.

To find the expected value E(X),

multiply X and P(X) for each case,

and add the products.

So

E(X) = (-1)⋅[1/4] + 1⋅[1/4] + 2⋅[1/3] + 3⋅[1/6].

Cancel (-1)⋅[1/4] and +1⋅[1/4].

2⋅[1/3] = 2/3

+3⋅[1/6] = +3/6

2/3 = 4/6

4/6 + 3/6 = 7/6

So E(X) = 7/6.

## Example 3

### Example

### Solution

You found that

a = 1/4 and E(X) = 7/6.

Then find E(X^{2}).

To find E(X^{2}),

first find X^{2}.

Square X values.

(-1)^{2} = 1

1^{2} = 1

2^{2} = 4

3^{2} = 9

Find E(X^{2}).

Multiply X^{2} and P(X) for each case,

and add the products.

So

E(X^{2}) = 1⋅[1/4] + 1⋅[1/4] + 4⋅[1/3] + 9⋅[1/6].

1⋅[1/4] = 1/4

+1⋅[1/4] = +1/4

+4⋅[1/3] = +4/3

+9⋅[1/6] = +9/6 = +3/2

The least common multiple of the denominators,

4, 3, 2,

is 12.

So, to add and subtract these fractions,

change the denominators to 12.

[1/4]⋅[3/3] = 3/12

+[1/4]⋅[3/3] = +3/12

+[4/3]⋅[4/4] = +16/12

+[3/2]⋅[6/6] = +18/12

3 + 3 = 6

16 + 18 = 34

6 + 34 = 40

Reduce 40 to, 40/4, 10

and reduce 12 to, 12/4, 3.

E(X) = 7/6

E(X^{2}) = 10/3

So

V(X) = 10/3 - (7/6)^{2}.

-(7/6)^{2}

= -[7^{2}/6^{2}]

= -[49/36]

10/3 = 120/36

120 - 49 = 71

So V(X) = 71/36.

## Example 4

### Example

### Solution

You found that

V(X) = 71/36.

Then σ(X) = √71/36.

√71/36 = √71/√36

Divide Radicals

√36

= √6^{2}

= 6

Square Root

So σ(X) = √71/6.