Ch. 5 Stats

Random variable

represented by X, has a single numerical value, determined by chance, for each outcome of a procedure.

Probability Distribution

A description that gives the probability for each value of the random variable. It is often expressed in the format of a table, formula, or graph.

Discrete random variable

A collection of values that is finite or countable.

Continuous random variable

has infinitely many values, and the collection values is not countable.

Probability Distribution Requirements

1. There is a numerical random variable x and its values are associated with corresponding probabilities.
2.?P(x)=1 where x assumes all possible values.
3. 0?P(x)?1 for every individual value of the random variable x.

Probability histogram

Vertical scale shows probabilities instead of relative frequencies

Mean for a probability distribution

�=?[x*P(x)]

Variance for a probability distribution

?�=?[(x-�)�*P{x)]

Variance for a probability distribution for manual computations

?�=?(x�*P(x)]-��

Standard deviation for a probability distribution

?=??[x�*P(x)]-��)

Expected value

Of a discrete random variable x is denoted by E, and it is the mean value of the outcomes, so E=� and E can also be found by evaluating ?[x*P(x)]

Binomial Probability Distribution

1. The procedure has a fixed number of trials-a single observation
2. The trials must be independent. The outcome of any individual trial doesn't affect the probabilities in the other trials.
3. Each trial must have all outcomes classified into 2 categori

Notation for Binomial Probability

p=probability
q=probability of a failure
n=denotes the fixed number of trials
x=the specific number of successes in n trials, so x can be any whole number between 0 and n
p=the probability of success in one of the n trials
q=the probability of failure in

Binomial Probability Formula

P(x)= n!�(n-x)!x!
p^x
*q^n-x

Binomial Distributions

Mean=�=np
Variance= ?�=npq
Standard Deviation =?=?npq

Max usual value

mean+2standard deviation

Poisson Distribution

Discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences of the event in an interval.The interval can be time, distance, area, volume or some similar unit.

Poisson Probability Distribution

P(x)= �^x*e^-��x!
e=2.71828
�=mean number of occurrences of the event over the intervals

Requirements for the poisson distribution

1. The random variable x is the number of occurrences of an event over some interval
2. the occurrences must be random
3. the occurrences must be independent of each other
4. the occurrences must be uniformly distributed over the interval being used

Parameters of the poisson distribution

The mean is�
The standard deviation is ?=?�

Requirements for using the poisson distribution as an approximation to the binomial

1. n?100
2. np?10

Mean for poisson distribution as an approximation to the binomial

�=np