Download 4.2 Random Variables and Their Probability

4.2 Random Variables and
Their Probability distributions
Streamlining Probability:
Probability Distribution,
Expected Value and Standard
Deviation of Random Variable
Graphically and
Numerically Summarize a
Random Experiment
Principal vehicle by which we do this:
random variables
Random Variables
Definition:
A random variable is a numerical-valued
variable whose value is based on the outcome of
a random event.
Denoted by upper-case letters X, Y, etc.
2 types of random variables:
i) Discrete: possible outcomes are a set of separate values,
“the number of …”
ii) Continuous: possible outcomes are an infinite
continuum
Examples
1. X = payout by insurance company on an
iPhone6 damage protection policy
Possible values of X are x=$0, $250, $500
2. Y=score on 13th hole (par 5) at Augusta
National golf course for a randomly
selected golfer on day 1 of 2015 Masters
y=3, 4, 5, 6, 7
Random Variables and
Probability Distributions
A probability distribution lists the possible values of a
random variable and the probability that each value will
occur.
Random variables are
unknown chance
outcomes.
Probability distributions
tell us what is likely
to happen.
Data variables are
known outcomes.
Data distributions
tell us what happened.
Probability Distribution Of Payout by
Insurance Company on an iPhone6
Damage Protection Policy
Policy payouts based on estimates of
damaged/ruined cellphones.
x
0
250
500
p(x)
0.67
0.13
0.20
Probability
Histogram
iPhone6 Insurance Policy Payouts
0.8
0.7
0.67
0.6
0.5
0.4
0.3
0.2
0.2
0.13
0.1
0
0
250
500
Probability Distribution Of Score on
13th hole (par 5) at Augusta
National Golf Course on Day 1 of
2015 Masters
y
3
4
5
6
7
p(x)
0.040
0.414
0.465
0.051
0.030
Score on 13th Hole
0.5
Probability
Histogram
0.465
0.45
0.414
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.051
0.04
0.03
0
3
4
5
6
7
Probability distributionsdiscrete random variables
Requirements
1. 0  p(x)  1 for all values x of X
2. all x p(x) = 1
Probability distributionscontinuous random variables
2 types of random variables:
i) Discrete: possible outcomes are a set of
separate values, “the number of …”
ii) Continuous: possible outcomes are an
infinite continuum
Probability distribution graphs for continuous
random variables come in many shapes.
The shape depends on the probability distribution
of the continuous random variable that the graph
represents.
Example graphs of probability distribution
functions of continuous random variables
1.2
1
f(x)
0.8
0.6
f(x)
f(x)
0.4
0.2
0
0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 6 6.3 6.6 6.9 7.2 7.5 7.8 8.1 8.4 8.7 9 9.3 9.6 9.9
Probabilities:
area under
graph
P(a < X < b)
X
a
b
P(a < X < b) = area under the graph between a and b.
Probability distribution
function of continuous rv
0  p(x)  1
 p(x)=1
Think of p(x) as the area
of rectangle above x
The sum of all
the areas is 1
 graph 0
the total area under
the graph = 1
Total area
under curve
=1
x
Expected Value of a
Random Variable
A measure of the “middle” of the
values of a random variable
Score on 13th Hole
iPhone6 Insurance Policy Payouts
0.8
0.7
0.67
0.6
0.5
0.4
0.3
0.2
0.2
0.13
0.1
0
0
250
500
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.465
0.414
0.051
0.04
3
4
5
6
The mean of the probability distribution is
the expected value of X, denoted E(X)
E(X) is also denoted by the Greek letter µ
(mu)
0.03
7
Mean or
Expected
Value
x
0
250
500
p(x)
0.67
0.13
0.20
y
3
4
5
6
7
p(x)
0.040
0.414
0.465
0.051
0.030
k = the number of possible values of
random variable
E( X )   =
k
 x  P( X  x )
i
i
i=1
E(x)= µ = x1·p(x1) + x2·p(x2) + x3·p(x3) +
... + xk·p(xk)
Weighted mean
Sample Mean
Mean or
Expected
Value
X
=
n
X

i
i = 1
n
x +x +x +...+x
n
X= 1 2 3
n
1
1
1
1
= x + x + x +...+ x
n 1 n 2 n 3
n n
k = the number of outcomes
E ( x)   =
k
x
i
 P(X=x i )
i=1
µ = x1·p(x1) + x2·p(x2) + x3·p(x3) + ... +
xk·p(xk)
Weighted mean
Each outcome is weighted by its probability
Other Weighted Means
GPA A=4, B=3, C=2, D=1, F=0
Five 3-hour courses: 2 A's (6 hrs), 1 B (3 hrs), 2 C's (6 hrs)
GPA:
4*6  3*3  2*6
15

45
 3.0
15
Course grade: tests 40%, final exam 25%,
quizzes 25%, homework 10%
Your scores: tests - 83, final exam - 75, quizzes - 90, homework - 100
Course grade  (83  .40)  (75  .25)  (90  .25)  (100  .10)
 33.2  18.75  22.5  10  84.45
"Average" ticket prices
Mean or
Expected
Value
x
0
250
500
p(x)
0.67
0.13
0.20
y
3
4
5
6
7
p(x)
0.040
0.414
0.465
0.051
0.030
E( X )   =
k
x
i
 P(X=x i )
i=1
E(X)= µ =0(0.67)+250(0.13)+500(0.20)
=32.5 + 100 = 132.5
E(Y)= µ=3(.04)+4(0.414)+5(0.465)+6(0.051)+7(0.03)
=4.617 strokes
Score on 13th Hole
Mean or
Expected
Value
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.465
0.414
0.051
0.04
3
4
5
6
0.03
7
µ=4.617
E(Y)= =3(.04)+4(0.414)+5(0.465)+6(0.051)+7(0.03)
=4.617 strokes
Interpretation of E(X)
E(X) is a “long run” average.
The expected value of a random variable
is equal to the average value of the
random variable if the chance process was
repeated an infinite number of times. In
reality, if the chance process is continually
repeated, x will get closer to E(x) as you
observe more and more values of the
random variable x.
 So the probability distribution of X is:
Example
x
p(x)
0
1/8
1
3/8
2
3/8
3
1/8
Let X = number of heads in 3 tosses of a
fair coin.
E(x) (or μ ) is
E(x)
4
  x  p(x )
i
i
i 1
 (0  1 )  ( 1 3 )  (2  3 )  (3  1 )
8
8
8
8
 12  1.5
8
US Roulette Wheel
and Table
 The roulette wheel has
alternating black and
red slots numbered 1
through 36.
 There are also 2 green
slots numbered 0 and
00.
 A bet on any one of
the 38 numbers (1-36,
0, or 00) pays odds of
35:1; that is . . .
 If you bet $1 on the
winning number, you
receive $36, so your
winnings are $35
American Roulette 0 - 00
(The European version has
only one 0.)
US Roulette Wheel: Expected Value of a
$1 bet on a single number
Let x be your winnings resulting from a $1 bet
on a single number; x has 2 possible values
x
p(x)
-1
37/38
35
1/38
E(x)= -1(37/38)+35(1/38)= -.05
So on average the house wins 5 cents on every
such bet. A “fair” game would have E(x)=0.
The roulette wheels are spinning 24/7, winning
big $$ for the house, resulting in …
Standard Deviation of a
Random Variable
First center (expected value)
Now - spread
Standard Deviation of a
Random Variable
Measures how “spread out” the
random variable is
Summarizing data and probability
Data
Histogram
measure of the
center: sample mean
x
measure of spread:
sample standard
deviation s
statistics
Random variable
Probability Histogram
measure of the
center: population
mean 
measure of spread:
population standard
deviation s
parameters
Variance –
measure of
spread
Variance
n
s2 =
 (X
i
 X) 2
i=1
n-1
=
1805.703
= 53.1089
34
The deviations of the outcomes from the
mean of the probability distribution
xi - µ
Xi - X
s2 (sigma squared) is the variance of the
probability distribution
[the variance is also denoted Var(X)]
Variance –
measure of
spread
Variance
n
s2 =
 (X
i
 X) 2
i=1
n-1
=
1805.703
= 53.1089
34
Variance of random variable X
s 2 [or Var(X)] =
k
2
(
x


)
 P ( X = xi )
 i
i =1
Variance Var(X)
x
0
250
500
p(x)
0.67
0.13
0.20
k
Var ( X ) =
2
(
x


)
 P ( X = xi )
 i
i =1
Recall: µ = E(X)=132.5
Example
132.5
132.5
Var(X) = (x1-µ)2 · P(X=x1) + (x2-µ)2 · P(X=x2)
+ (x3-µ)2 · P(X=x3)
132.5
= (0-132.5)2 · 0.67 + (250-132.5)2 · 0.13
+ (500-132.5)2 · 0.20 = 40,568.75
P. 207, Handout 4.1, P. 4
Standard Deviation: of
More Interest then the
Variance
The population standard deviation, denoted by s or SD(X),
is the square root of the population variance Var(X)
s  s  or SD( X ) = Var ( X )
Standard Deviation
Standard
Deviation
Standard Deviation (s) =
Positive Square Root of the Variance
s =
s2
s2 = 40,568.75
s, or SD(X), is the standard deviation of the
random variable X
s [or SD(X)] = s
2
s (or SD( X )] = 40,568.75  201.42