Business Statistics Unit 2

Probability

The chance that something will happen

Experiment

An action that produces an outcome

Sample space

One or more outcomes of an experiment

Simple event

An event with exactly one outcome

Exhaustive events

Events that together include every possible outcome

Mutually exclusive events

Events that cannot occur at the same time

Classical

Based on equally likely outcomes

Empirical

Based on observed data or past results

Subjective

Based on personal judgment or experience

Odds

compare the chance of success with the chance of failure

Odds in favor equation

P(A)/1-P(A)

Independent events

One event does not affect the probability of the other.

Dependent events

One event affects the probability of the other

Independent events equation

P(A∩B)=P(A)×P(B)

conditional probability equation

P(A∣B)=P(A∩B)​/P(B)

Discrete random variable

Takes countable, distinct values

use a probability mass function

Continuous random variable

Can take any value within an interval.

use a probability density function

Probability mass function (PMF)

Tells you the probability of each possible value of a discrete random variable

0≤P(X=x)≤1, ∑P(X=xi​)=1

1. Every probability must be between 0 and 1.

2. All probabilities must add up to 1.

probability distribution

gives the probability of an individual value

cumulative distribution function (CDF)

Gives the probability of a value less than or equal to a specified number

F(x)=P(X≤x)

Contingency tables

organizes counts or probabilities into rows and columns

To find a missing value:

Row or column total − Sum of the known values

Consumer type

How they make decisions

Risk-neutral

Focuses on expected gains and ignores risk itself

Risk-averse

Prefers less risk and generally requires compensation to accept additional risk

Risk-loving

Prefers risk and may accept a gamble even when its expected gain is negative

Binomial distribution

Counts the number of successes in a fixed number of trials.

Include:

• Fixed number of trials n.

• Two possible outcomes per trial.

• Constant probability of success pp.

• Independent trials.

Binomial distribution formula

P(X=x)=(n/x​)p^x(1−p)^n−x

Poisson distribution

Counts the number of occurrences in a specified interval of time or space.

Poisson distribution formula

P(X=x)=e^−λ*λ^x/x!​

What does a probability density function(PDF) describe?

the shape of the distribution of a continuous random varriable

What is the first property of a PDF?

f(x) is greater than or equal to 0. The density cannot be negative

What is the total area under a PDF curve?

1 or 100%

What happens to the tails of a normal PDF as x approaches positive infinity?

They approach the x-axis but never touch it

What is the probability of a continuous random variable equaling one exact value?

P(X=a)=0

How do you calculate probabilities for a continuous random varriable?

Find the area under the PDF curve over the interval

What is the formula for a cumulative distribution function?

F(x)=P(X less than or equal to x)

What does a CDF represent?

The probability that X is less than or equal to x

the area under the PDF to the left of x.

If F(10) = 0.70, what does this mean?

There is a 70% probability that X is less than or equal to 10

What is a continuous uniform distribution?

a distribution with constant density over the interval [a,b]

What is the formula for the height of an uniform density curve?

f(x)= 1/(b-a)

What is the formula for probability over an interval in a uniform distribution?

interval length / total interval length

If X is uniform from 2 to 8, what is P( 3 less than or equal to X less than or equal to 5)?

(5-3)/(8-2)= 2/6, .333, or 33.33%

What is the shape of a normal distribution?

A symmetric, bell shaped curve

Where is the mean located in a normal distribution?

At the center of the curve

In a normal distribution, how are the mean, median, and mode related?

they are all equal

What is the total area under a normal curve? ,

1, or 100%

What percentage of a normal distribution lies below the mean?

50% or .50

What percentage of a normal distribution lies above the mean? ,

50%, or .50

What two parameters determine a normal distribution?

the mean (μ) and the standard deviation (σ) (or the variance (σ²)

What does a z score tell you?

How many standard deviations a value is above or below the mean

What is the formula for a z-score?

z =(x-μ)/σ

What does a positive z score mean?

The value is above the mean

What does a negative z score mean?

the value is below the mean

What does a z core of zero mean?

the value equals the mean

The mean is 100, the standard deviation is 15, and x =130. What is z?

z = (130 -100)/15=2. The value is 2 standard deviations above the mean.

What is the empirical rule for 1 standard deviation?

Approximately 68% of observations fall between (μ) - (σ) and (μ) + (σ)

What is the empirical rule for 2 standard deviations?

Approximately 95% of observations fall between (μ) - (2σ) and (μ) + (2σ)

What is the empirical rule for 3 standard deviations?

Approximately 99.7% of observations fall between (μ) - (3σ) and (μ) + (3σ)

The mean is 50 and the standard deviation is 10. What is the range within 1 standard deviation?

40 to 60, containing approximately 68% of observations

The mean is 50 and the standard deviation is 10. What is the range within 2 standard deviations?

30 to 70, containing approximately 95% of observations

The mean is 50 and the standard deviation is 10. What is the range within 3 standard deviations?

20 to 80, containing approximately 99.7% of observations

If x is below the mean, what is P(X > x)?

Greater than .50

If x is below the mean, what is P(X < x)?

Less than .50

If x is above the mean, what is P(X > x)? l

less than .50

If x is above the mean, what is P(X < x)? g

greater than .50

If μ = 10, which is larger: P(X>8), or P(X<8)

P(X>8) is larger because 8 is below the mean

If x equals the mean, what are P(X<x) and P(X>x)?

both equal .50