y
0
=
y
(
x
0
)
y
1
=
y
0
+
hF
(
x
0
, y
0
)
y
2
=
y
1
+
hF
(
x
1
, y
1
)
.
.
.
y
n
=
y
n

1
+
hF
(
x
n

1
, y
n

1
)
Then,
y
n
gives an approximation of the true value
y
(
b
) of teh solution to the initial value problem at
x
=
b
.
*
Exercises: p.640 115
•
Chapter 10 Probability and Calculus
–
10.1 Probability Distributions of Random Variables
*
Terms: experiment, outcomes, sample points, sample space, event, probability of the event, finite discrete
random variable, discrete probability function, histogram, continuous random variable, probability density
function, exponential density function, exponentially distributed, joint probability density function
*
Probabilty Density Function: A probability density function of a r.v.
X
in an interval
I
, where
I
may be
bounded or unbounded, is a nonnegative function
f
having the property that the total area of the region
under the graph of
f
in the interval
I
is equal to 1. The probability that an observed value of the random
variable
X
lies in the interval [
a, b
] is given by
P
(
a
≤
X
≤
b
) =
Z
b
a
f
(
x
)
dx
*
For a continuous random variable
X
,
P
(
a
≤
X
≤
b
) =
P
(
a < X
≤
b
) =
P
(
a
≤
X < b
) =
P
(
a < X < b
).
*
A joint probability density function of the random variables
X
and
Y
on a region
D
is a nonnegative
function
f
(
x, y
) having the property
Z
D
Z
f
(
x, y
)
dA
= 1
The probability that the observed values of teh random variables
X
and
Y
lie in a region
R
⊂
D
is given
by
P
[(
X, Y
) in
R
] =
Z
R
Z
f
(
x, y
)
dA
*
Exercises: pp.654655 142
–
10.2 Expected Value and Standard Deviation
*
Terms: expected value, variance, standard deviation
*
Expected Value of a Discrete Random Variable
X
: Let
X
denote a random variable that assumes the
values
x
1
, x
2
, . . . , x
n
with associated probabilities
p
1
, p
2
, . . . , p
n
, respectively. Then the expected value of
X, E
[
X
], is given by
E
[
X
] =
x
1
p
1
+
x
2
p
+ 2 +
· · ·
+
x
n
p
n
*
Expected Value of a Continuous Random Variable: Suppose the function
f
defined on the interval [
a, b
]
is the probability density function associated with a continuous random variable
X
. Then, the expected
value of
X
is
E
[
X
] =
Z
b
a
xf
(
x
)
dx
*
Variance of a Discrete Random Variable: Let
X
be the discrete random variable two points above, and
let
X
have expected value
E
[
X
] =
μ
. Then the variance of the random variable
X
is
Var(
X
) =
p
1
(
x
1

μ
)
2
+
p
2
(
x
2

μ
)
2
+
· · ·
p
n
(
x
n

μ
)
2
*
Variance of a Continuous Random Variable: Let
X
be a continuous random variable with probability
density funciton
f
(
x
) on [
a, b
] and expected value
E
[
X
] =
μ
. Then the variance of
X
is
Var(
X
) =
Z
b
a
(
x

μ
)
2
f
(
x
)
dx
9