Download Learning Objectives Random Variable Probability Distribution

6.1.notebook
August 06, 2015
Learning Objectives
6.1 Summarize Possible Outcomes and their Probabilities
• Random variable
• Probability distributions for
discrete random variables
• Mean of a probability
distribution
• Summarizing spread of
probability distribution
• Probability distribution for
continuous random variables
cdn.information­management.com
Aug 8­9:27 AM
Random Variable
www.economicpopulist.org
Aug 8­9:27 AM
Probability Distribution
• Letters near end of alphabet, x,
symbolize
• Variables
• Particular value of random
variable
• Capital letter, X, refer to
random variable itself
Ex: Flip a coin three times
• X = # of heads in 3 flips;
defines random variable
• x = 2; represents value of
random variable
The probability
distribution of random
variable specifies its
possible values and
their probabilities
Note: Randomness of
variable allows us to
give probabilities for
outcomes
Aug 8­9:27 AM
Probability Distribution of Discrete Random
Variable
psy2.ucsd.edu
• A discrete random
variable X has
separate value
(0,1,2,…) outcomes
• For each x,
probability
distribution assigns P
(x):
• Between 0 and 1
• Sum of all is 1
Aug 8­9:27 AM
www.anticsland.com
Aug 8­9:27 AM
Example
• What is the estimated probability of at least three home runs?
P(3)+P(4)+P(5)=0.13+0.03+0.01=0.17
8
Aug 8­9:27 AM
1
6.1.notebook
August 06, 2015
The Mean of a Discrete Probability
Distribution
Expected Value of X
• The mean of a probability
distribution, µ, for a discrete
random variable is
• Mean of probability
distribution is also
expected value of X
• Reflects not single
observation but what
we expect for average
of lots of observations
• Often NOT a possible
outcome
• Mean is a weighted
average; more likely values
receive greater weight, P(x)
cnfolio.com
9
Aug 8­9:27 AM
Aug 8­9:27 AM
The Standard Deviation of a Probability
Distribution
Learning Objective 3:
Example
• Find the mean of this probability distribution.
The standard deviation of a
probability distribution, σ,
measures spread
• Larger σ corresponds to
greater spread
• σ describes how far random variable falls, on average, from mean
The mean:
= 0(0.23) + 1(0.38) + 2(0.22) + 3(0.13) + 4(0.03) + 5(0.01) = 1.38
rchsbowman.files.wordpress.com
11
Aug 8­9:27 AM
Continuous Random Variable
• Has infinite continuum
of possible values in an
interval
• Measures: time, age,
size, height, weight, …
• Continuous variables
are rounded to discrete
values
Aug 8­9:27 AM
Aug 8­9:27 AM
Probability Distribution of a Continuous
Random Variable
A continuous random variable has possible
values that form an interval and a
probability distribution that is specified by
a curve.
• Each interval has probability between 0
and 1.
• The interval containing all possible values
has probability equal to 1.
Aug 8­9:27 AM
2
6.1.notebook
August 06, 2015
Aug 8­9:27 AM
Aug 14­3:23 PM
3