Transcript Chapter 7 Discrete Distributions Random Variable

Chapter 7
Discrete Distributions
Random Variable • A numerical variable
whose value depends
on the outcome of a
chance experiment
Two types:
• Discrete – count of some
random variable
• Continuous – measure of
some random variable
Discrete Probability Distribution
• Gives the values associated
with each possible x value
• Usually displayed in a table,
but can be displayed with a
histogram or formula
Properties for a discrete
probability distribution
1) For every possible x value,
0 < P(x) < 1.
2) For all values of x,
S P(x) = 1.
Suppose you toss 3 coins & record
the number of heads.
The random variable X defined as ...
The number of heads tossed
Create a probability distribution.
X
P(X)
0
.125
1
.375
2
.375
3
.125
Create a probability histogram.
Let x be the number of courses for which
a randomly selected student at a certain
university is registered.
X
1 2 3
P(X).02 .03 .09
4 5 6 7
? .40 .16 .05
Why
does
this
not
start
at
zero?
.25
P(x = 4) =
P(x < 4) = .14
P(x < 4) = .39
P(x > 5) = .61
What is the probability that the student
is registered for at least five courses?
Formulas for mean & variance

  x
x 
x
2
xi p
i
i
 x  p
2
i
Found on formula card!
Let x be the number of courses for which
a randomly selected student at a certain
university is registered.
X
1
2
3
4
5
6
7
P(X).02 .03 .09 .25 .40 .16 .05
What is the mean and standard
deviations of this distribution?
  4.66
&
 = 1.2018
Here’s a game:
If a player rolls two dice
and gets aAsum
of 2is or
12,
fair game
one where
the cost to play EQUALS
he wins $20.theIf
he
gets
a
expected value!
0 $5. 5The cost
20
7,X he wins
to
P(X)
7/9
1/6
1/18
roll the dice one time is
$3. Is this game fair?
NO, since  = $1.944 which is less
than it cost to play ($3).
Linear function of a random
variable
The mean is changed
If xbyis addition
a random&variable and a and b
aremultiplication!
numerical constants, then the
random variable y is defined by
The standard deviation is
y  a  bx
ONLY changed by
• and
multiplication!
 y   a bx  a  b x
2
y

2
a bx

2 2
b x
or  y  b  x
Let x be the number of gallons
required to fill a propane tank.
Suppose that the mean and
standard deviation is 318 gal. and 42
gal., respectively. The company is
considering the pricing model of a
service charge of $50 plus $1.80
per gallon. Let y be the random
variable of the amount billed. What
is the mean and standard deviation
for the amount billed?
 = $622.40 &  = $75.60
JustLinear
add or subtract
combinations
the means!
If y  x1  x 2  ...  xn then
 y   x 1   x 2  ...   xn
y 
2
x 1
2
 x 2
 ...
2
  xn
If independent, always add
the variances!
A nationwide standardized exam consists of a
multiple choice section and a free response
section. For each section, the mean and
standard deviation are reported to be
mean
SD
MC
38
6
FR
30
7
If the test score is computed by adding the
multiple choice and free response, then what is
the mean and standard deviation of the test?
  68
&
 = 9.2195