Terms
What kind of variable has a single numerical value which is determined by chance for each outcome of a procedure?
(Denoted using an x)
Random Variable
Calculate the max and min usual value if the mean is 5 and the standard deviation is 2.
In other words, find the range of possible values that would be considered "usual".
Max: 5 + 2(2) = 9
Min: 5 - 2(2) = 1
Between 1 and 9 would be usual
What is the variable used to denote the probability of of failure in one trial?
q
Parameters for Binomial Distributions:
1. Mean = np
2. Variance = npx
3. Standard Deviation = the square root of npq
2. Variance = (npq)
Not = npx
Do expected values have to be whole numbers?
No...do not round them!
What kind of random variable has a collection of values that is finite or countable?
Discrete Random Variable
Calculate the variance and standard deviation if n = 6 and p = 0.45
Variance: npq = 6(0.45)(1-0.45) or 6(0.45)(0.55)
= 1.485
Standard Deviation = the square root of (npq)
= the square root of (1.485)
= 1.22 (rounded)
What does the equation 1 - p result in or equal?
q
Probability of a failure
Binomial Probability Distributions:
1. P(F) = p
2. P(F) = 1 - p
3. P(F) = q
1. P(F) = (q)
Not = p
How is this word pronounced? Write it out.
"Poisson"
Pwah saan
What is denoted using E, and is the mean value of the outcomes of a discrete random variable x?
Expected Value
Use the makeshift table to find the mean of a probability distribution.
x l P(x)
0 l 0.61
1 l 0.15
2 l 0.24
Mean = sum of (x times P(x))
sum of (x times P(x)) = 0.00 + 0.15 + 0.48
Mean = 0.63
What is the variable used to denote a specific number of successes in a certain number of trials?
x
Range Rule of Thumb:
1. Most values should fall within 2 standard deviations of the mean
2. This helps us identify "unusual" values
3. Max and min usual values can be found by taking the mean and adding or subtracting two times the variance, respectively.
3. Max and min usual values can be found by taking the mean and adding or subtracting two times the (standard deviation), respectively
Not variance
What page was that super useful table or list of binomial probabilities on in our textbook?
Page 723
(You probably want to have that page bookmarked for your exam ;)
What is the term for a description that gives the probability for each value of the random variable?
(It is often expressed as a table, formula, or graph)
Probability Distribution
Find the probability of exactly 4 successes happening in a Poisson Probability when the mean is 3.
Poisson Formula: P(x) = (mean raised to the x, times e raised to the negative mean) / x factorial
P(4) = (3 raised to the 4 times e raised to the -3) / 4 factorial
P(4) = (81 * 0.049787) / 4*3*2*1
P(4) = 4.033 / 24
P(4) = 0.168
What does the variable p stand for or denote?
Be specific...list the whole description
The probability of success in one of the n trials.
Probability Distribution Requirements
1. A numerical random variable "x"
2. The sum of all probabilities is 100
3. Each probability value must be between 0 and 1 inclusive
2. The sum of all probabilities is (1)
Not (100)
Expected value is the same as what in terms of probability distributions?
The mean or the sum of (x times P(x))
What results from a procedure that meets all four of these requirements?
1. Fixed number of trials
2. Trials are independent
3. Two possible outcomes (success or fail typically)
4. Probability remains the same in all trials
Binomial Probability Distribution
Calculate the standard deviation for a probability distribution if the sum of (x squared times P(x)) is 8 and the mean is 2.
Standard deviation is the square root of [(x squared times P(x)) - the mean squared]
The square root of [(8) - 2 squared] is the square root of (8 - 4) or the square root of 4... which is 2.
2
FINAL JEOPARDY
BPF vs SPF
Requirements for Poisson Distribution:
1. Random variable x is the occurrences over some interval
2. Occurrences are random
3. Occurrences are independent
4. Occurrences are distributed in a normal bell-shape
4. Occurrences are (uniformly distributed)
Not normal bell-shape
What is the approximation or value we are going to use for e in our Poisson probability distributions?
e = 2.71828