Transcript discrete

4.3
RANDOM VARIABLES
Discrete and continuous random variables
Normal distributions as probability distributions
4.4
MEANS AND VARIANCES OF RVs
Rules for means of random variables
Rules for variances of random variables
The law of large numbers
Discrete random variables
 A discrete
random variable can take on
only a certain number of possible
values.
•
Discrete random variables
The
probability distribution for a discrete
random variable can be given by a list of
the possible values, and the probability of
each one.
value of X
probability
2
3
4
5
6
7
8
9
10
11
12
.028 .056 .083 .111 .139 .167 .139 .111 .083 .056 .028
Discrete random variables
“How many heads will I get if I toss two coins?”
sample space = {TT, TH, HT, HH}
0.5
Number
of heads
0
1
2
0.4
Probability
.25
.50
.25
0.3
0.2
0.1
0
0
1
2
Continuous random variables
Continuous random variables can take all
possible values in a particular interval.
The probability distribution of a continuous
random variable is described by a density
curve.
The probability that the random variable falls
within a certain interval is given by the area
under the curve in that interval.
Normal distributions as
probability distributions
“What is the probability that a woman
selected at random is 67 inches tall or taller?”
The heights of adult American women are
approximately normally distributed with m=64.5
inches and s=2.5 inches.
z = (67-64.5) / 2.5 = +1.00
Table A tells us that about 16% of a normal curve
lies to the right of z = +1.00, so the probability is
about .16.
4.4 MEANS and VARIANCES
of RANDOM VARIABLES
Means of random variables
The mean of a discrete random variable
is equal to the sum of each possible
value multiplied by its probability.
mX  x1p1  x 2p 2 
 x kpk
For example, the mean of the number of
heads we see when we toss two coins is
mX  (0).25  (1).50  (2).25  1
Means of random variables
The mean of a continuous random
variable is the point at which the density
curve would balance.
The mean of any random variable is
also known as its expected value.
Statistical estimation
The law of large numbers says that if you
want an extremely accurate estimate of m, all
you have to do is to draw an extremely large
random sample, and your sample mean will
be acceptably close to m.
In other words, “the average results of many
independent observations are stable and
predictable.”
Rules for means
1 The expected value of a linear transformation
is the linear transformation of the expected
value:
m(a  bX)  a  bmX
2 The expected value of a sum is the sum of
the expected values:
m(X  Y)  mX  mY
The variance of a discrete
random variable
The variance of a discrete random variable is
the sum of each possible squared deviation
multiplied by its probability:
s X  (x1 - mX ) p1   (x k - mX ) p k
2
2
2
For example, the variance of the number of heads
we see when we toss two coins is
ó
2
X
 (0 - 1) .25  (1  1) .50  (2 - 1) .25  0.5
2
ó X  0.707
2
2
Rules for variances
1 The variance of a linear transformation is the original
variance times the square of the multiplier:
ó
2
(a  bX)
b ó
2
2
X
2 The variance of a sum is the sum of the variances
(if X and Y are independent):
ó
2
(X  Y)
ó
2
X
ó
2
Y
The variance of a difference is the sum of the variances
(if X and Y are independent):
ó
2
(X  Y)
ó
2
X
ó
2
Y