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Transcript
5.3 Random Variables
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1
Random Variable
Discrete Random Variables
Continuous Random Variables
Normal Distributions as Probability Distributions
Random Variables
A random variable is a variable whose values are numerical
outcomes of a random experiment. That is, we consider all the
outcomes in a sample space S and then associate a number with
each outcome
A random variable takes numerical values that describe the
outcomes of some chance process.
The probability distribution of a random variable gives its
possible values and their probabilities. The probability histogram is
a graph showing this probability distribution…
Example: Consider tossing a fair coin 3 times.
Define X = the number of heads obtained in 3
tosses
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X 2= 3: HHH
X Values
0
1
2
3
Probability
1/8
3/8
3/8
1/8
• The figure below shows how to get the probability
distribution of X – use a tree diagram! Each outcome
has prob=1/16 (HINT: use the “and” rule to show
this), and then use the “or” rule to show that P(X=1)
= P(TTTH or TTHT or THTT or HTTT) etc…)
Discrete Random Variable
There are two main types of random variables: discrete and continuous. If we
can find a way to list all possible outcomes for a random variable and assign
probabilities to each one, we have a discrete random variable.
A discrete random variable X takes on a fixed set of possible values with
gaps between. The probability distribution of a discrete random variable X
lists the values xi and their probabilities pi:
Values of X:
P(X)
:
x1
p1
x2
p2
x3
p3
…
…
The probabilities pi must satisfy two requirements:
1. Every probability pi is a number between 0 and 1.
2. The sum of the probabilities is 1.
To find the probability of any event regarding X, add the probabilities pi of
the particular values xi that make up the event.
4
Continuous Random Variable
Discrete random variables commonly arise from situations that involve
counting something. Situations that involve measuring something often result
in a continuous random variable.
A continuous random variable Y takes on its values in an interval of
numbers. The probability distribution of Y is described by a density
curve. The probability of any event regarding Y is the area under the
density curve and above the values of Y that make up the event.
5
Continuous Probability Models
Suppose the random expt. is “choose a number at random between 0 and 1”.
We cannot assign probabilities to each individual value because there is an
infinite interval of possible values, so we model this with a density curve (that
is flat) that gives equal weight to all the numbers between 0 and 1.
This continuous probability model assigns probabilities as
areas under the flat, uniform density curve. The area under
the curve and above any range of values is the probability of an
outcome in that range. The total area under the curve equals 1.
6
Example: Find the probability of
getting a random number that is
less than or equal to 0.5 OR
greater than 0.8.
P(X ≤ 0.5 or X > 0.8)
= P(X ≤ 0.5) + P(X > 0.8)
= 0.5 + 0.2
= 0.7
Uniform
distribution
Normal Probability Models
There are many situations where the density curve used to assign
probabilities to intervals of outcomes is Normal.
Normal distributions can be thought of as probability models:
Probabilities can be assigned to intervals of outcomes using the Standard
Normal probabilities in Table A.
We standardize normal data by calculating z-scores so that any Normal
curve N(m,s) can be transformed into the standard Normal curve N(0,1).
z
7
(x  m)
s
z
Normal Probability Models
Women’s heights are Normally distributed with mean 64.5 and standard
deviation 2.5 in. What is the probability, if we pick one woman at random,
that her height will be between 68 and 70 inches P(68 < X < 70)?
Because the woman is selected at random, X is a random variable.
z
(x  m)
s
As before, we calculate the zscores for 68 and 70.
For x = 68",
z
(68  64.5)
 1.4
2.5
For x = 70",
z
(70  64.5)
 2.2
2.5
8

5.4 Means and Variances
of Random Variables
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The Mean of a Random Variable
The Law of Large Numbers
The Variance of a Random Variable
HW problems for sections 5.3 and 5.4
The Mean (Expected Value) of a Random Variable
The mean of any discrete random variable is an average of the
possible outcomes, with each outcome weighted by its
probability. This reflects the fact that all outcomes may not be
equally likely. The mean is also called the expected value.
Mean of a Discrete Random Variable
Suppose that X is a discrete random variable whose probability
distribution is
Value:
x1 x2 x3 …
Probability: p1 p2 p3 …
To find the mean (expected value) of X, multiply each possible value
by its probability, then add all the products – so it’s a weighted sum of
the r.v.’s values, the weights being the probabilities.
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μx  E ( X )  x1 p1  x2 p2  x3 p3  ...
  xi pi
• We’ve already discussed the mean of a density
curve as being the “balance point” of the curve… For
a discrete r.v., we’ll compute the mean (or expected
value) as a weighted average of the values of X, the
weights being the corresponding probabilities and in
either case (discrete or continuous), the
interpretation of the mean is as the long-run
average value of X (in a large number of repetitions
of the experiment giving rise to X).
• Example: The mean # of Hs in 4 tosses of a fair coin is
computed as: (1/16)*0 + (4/16)*1 + (6/16)*2 +
(4/16)*3 + (1/16)*4 = (32/16) = 2.
• What is your mean “winnings” when you play the
“Pick 3” lottery? What does it mean?
The Law of Large Numbers
Law of Large Numbers: Essentially states that if you sample
from a population with mean = m, then the sample mean (xbar) will approximate m for large enough sample sizes. Or
that m can be thought of as the expected value (long-run
average value) of many independent observations on the
variable.
Draw independent observations at random from any
population with finite mean µ. Decide how accurately you
would like to estimate µ. The law of large numbers says that as
you increase the number of observations, the sample mean of
the observed values gets closer and closer to the mean µ of the
population.
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Variance of a Random Variable
Since we use the mean as the measure of center for a
discrete random variable, we’ll use the standard deviation as
our measure of spread. The definition of the variance of a
random variable is similar to the definition of the variance
for a set of quantitative data.
Variance of a Discrete Random Variable
Suppose that X is a discrete random variable whose probability
distribution is:
Value:
x1 x2 x3 …
Probability: p1 p2 p3 …
and that µX is the mean of X. The variance of X is:
Var(X)  s X2  (x1  m X ) 2 p1  (x 2  m X ) 2 p2  (x 3  m X ) 2 p3  ...
  (x i  m X ) 2 pi
To get the standard deviation of a random variable, take the square root
of the variance.
13
HW: Read sections 5.3 and 5.4 (up through p. 273, Law of Large
Numbers)… Don’t worry much about the computation of the
variance of a discrete random variable…
Work on Exercises # 5.43, 5.48-5.55