Chapter 6: Probability Distributions. Section 6.1: How Can We Summarize Possible Outcomes and Their Probabilities?. Learning Objectives. Random variable Probability distributions for discrete random variables Mean of a probability distribution
Section 6.1: How Can We Summarize Possible Outcomes and Their Probabilities?
Example: Flip a coin three times
Note: It is the randomness of the variable that allows us to specify probabilities for the outcomes
where the sum is taken over all possible values of x.
= 0(0.23) + 1(0.38) + 2(0.22) + 3(0.13) + 4(0.03) + 5(0.01) = 1.38
The standard deviation of a probability distribution, denoted by the parameter, σ, measures its spread.
Section 6.2: How Can We Find Probabilities for Bell-Shaped Distributions?
The normal distribution is symmetric, bell-shaped and characterized by its mean µ and standard deviation .
[ µ = 65 3.5 ]
[ µ 2 = 65 2(3.5) = 65 7 ]
[ µ 3 = 65 3(3.5) = 65 10.5 ]
(by 68-95-99.7 Rule)
65 68.5 (height values)
? = 84%Learning Objective 2:Example : 68-95-99.7% Rule
Table A enables us to find normal probabilities
To use the table:
TI Calculator = Normcdf(-1e99,1.43,0,1)= .9236
TI Calculator = Normcdf(1.43,1e99,0,1)= 0.0764
TI Calculator = Normcdf(-1.43,1.43,0,1)= .8472
To calculate the cumulative probability
random variable falls at least 1.96
standard deviations below the
mean is 0.025.
random variable takes a value no more
than 1.96 standard deviations above
the mean is 0.975.
What is z? Invnorm(.975,0,1)=1.96
What is z? Invnorm(1-.881,0,1)=-1.18
What is z? Invnorm(.119,0,1)= -1.18
Z-scores can be used to compare observations from different normal distributions
=100 and 30 on the ACT which has =21.0 and
=4.7. On which test did you perform better?
Section 6.3: How Can We Find Probabilities When Each Observation Has Two Possible Outcomes?
Rules for factorials:
If John Doe does not actually have ESP and is actually guessing the number, what is the probability that he’d make a correct guess on two of the three trials?