Statistics

A probability distribution is a description that gives the probability for each value of the random variable which can be a table, graph, or formula and describes what will probably happen instead of what actually did happen !!!!

Every probability distribution must satisfy two requirements:

1. The probability of each event must be between 0 and 1

0≤P(x)≤1    This means no negative probabilities, and a probability cannot exceed 1.

2. The sum of all probabilities must equal 1

∑P(x)=1

The mean of a discrete random variable is the theoretical mean outcomes for infinitely many trials.

We can call this the expected value, in the sense that it is the average value that we would expect if the trials could continue indefinitely

Formula:

The discrete uniform probability distribution is the simplest example of a discrete probability distribution.

It is defined as: If a random variable, x, follows the discrete uniform distribution on the interval [a, a+1,...,b],if it may attain each of these values with equal probability, given by a formula:

where: n = the number of values the random variable may assume.

A binomial probability distribution results from a procedure that meets all of the following:

1. Fixed number of trials

2. Independent trials :

Outcome of any particular trial doesn’t affect the probabilities in other trials

3. Trial outcomes can be classified into two categories Usually success or failure

4. Probability of success remains the same in all trials

P: Probability of success in one of the n trials

q: Probability of a fail in one of the n trials

 P(x): Probability of getting exactly x successes among the n trials

• n = number of trials

• p = probability of success

• q= probability of failure = 1−p

If you flip a coin 10 times, and the chance of heads is 0.5:

• n=10n 

• p=0.5p 

• q=1−0.5=0.5

Used to describe the behaviour of rare events with small probabilities

It is a discrete probability distribution that applies to occurrences of some event over a specified interval

x is the number of occurrences of the event in an interval

The interval can be time, distance, area, volume, or some similar unit

Requirements:

1. Random variable x is the number of occurrences of an event over a specified interval

2. Occurrences must be random

3. Occurrences must be independent of each other

4. Occurrences must be uniformly distributed over the interval being used

Uniformly distributed—a number of values are equally likely to be observed

• μ = mean number of occurrences in the given interval, x = number of actual occurrences, e= Euler’s number ≈ 2.71828

If the average number of emails received per hour is 4 (μ = 4):

• To find the probability of receiving exactly 2 emails in an hour:

•