Probability distribution is the distribution of probabilities of all possible outcomes of an experiment. The distribution can involve outcomes with equal or different likelihoods.


Random variables

Notations that are used when solving probability distribution problems include X and x.
X is a random variable that can have any of a set of different values, while its corresponding lowercase letter x represents its individual values.

Note: X is a variable therefore it is written in words and x is written in numbers because it represents the numerical value of X.


Example:
X = number of times a head is flipped in one coin toss
x = {0, 1}


Types of random variables:


1. Discrete Random Variable
- possible values are integers
- consists of distinct values that are separate from each other
- there's always a finite number of outcomes

2. Continuous Random Variable
- values are continuous in nature (such as human height) and can be in decimal or fractional form
- X = possible values, x = any real numbers

Example:
Classify the following random variables as discrete or continuous.
1. length of time you stay in a class
2. number of classes you attend in a day


Solution
1. continuous
2. discrete


Uniform Distribution is the distribution of probabilities with equally likely outcomes.

Uniform random variable with 'n' outcomes

Example 1:
Determine whether the following experiments has a uniform probability
1. guessing a person's age
2. rolling a number on a dice


Solution
1. not uniform - some values will have a higher probability of being selected (we take physical appearance and other exterior influences into account thus affecting our guess)
2. uniform - a number on a die can be any number that only appears once.

Example 2: Uniform Probability Distribution
Determine the probability distribution for rolling a number on a dice.

Solution
Each number on the dice have an equal probability of being rolled of the six numbers therefore each probability is 1/6.




Note: Probabilities of a complementary event must add up to 1.



Expected Value is the average of the outcomes.

"Expected Value" or "Expectation"

Example:
Given the following probability distribution, determine the expected value.

x	P(x)
2	0.4
4	0.1
6	0.5

Solution
E(X) = 2(0.4) + 4(0.1) + 6(0.5)
= 0.8 + 0.4 + 3
= 4.2

∴ The expected value is 4.2.

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7.1 EXAMPLE


1.
a)Determine the probability distribution for the sum rolled with two dice.
b)Determine the expected sum of two dice.


Solution

Sums of two dice



a) X = sum of two dice


































b)


∴ The expected value is 7.


KEY CONCEPTS

Experiment obeys: all outcomes equally probable
Random variable: outcome
Example: tossing a fair die (n=6)




References

Wgman, Diane. Mathematics of Data Management. Toronto: McGraw-Hill Ryerson, 2002.