expectation random variables graphs and histograms expected value
Post on 20-Dec-2015
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Expectation
Random Variables
Graphs and Histograms
Expected Value
Random Variables
A random variable is a rule that assigns a numerical value to each outcome of an experiment. We will classify random variables as either
Finite discrete – if it can take on only finitely many possible values.
Infinite discrete – infinitely many values that can be arranged in a sequence.
Continuous – if its possible values form an entire interval of numbers
Example One
Suppose that we toss a fair coin three times. Let the (finite discrete) random variable X denote the number of heads that occur in three tosses. Then
Sample
Pt. Value of X
Sample Pt. Value of X
HHH 3 HTT 1
HHT 2 THT 1
HTH 2 TTH 1
THH 2 TTT 0
Example Two
Suppose that we toss a coin repeatedly until a head occurs. Let the (infinite discrete) random variable Y denote the number of trials.
Sample
Pt. Value of Y
Sample Pt. Value of Y
H 1 TTTTH 5
TH 2 TTTTTH 6
TTH 3 TTTTTTH 7
TTTH 4
Example Three
A biologist records the length of life (in hours) of a fruit fly. Let the (continuous) random variable Z denote the number of hours recorded. If we assume for simplicity, that time can be recorded with perfect accuracy, then the value of Z can take on any nonnegative real number.
Graphs and Histograms
Given a random variable X, we will be interested in the probability that X takes on a particular real value x, symbolically we write
pX(x) = P(X = x)
pX(x) is referred to as the probability function of the random variable X.
Geometric Representation
Consider Example Two where a coin is tossed three times. From the given table we see thatp(0)= P(X= 0)= 1/8
p(1)= P(X= 1)= 3/8
p(2)= P(X= 2)= 3/8
p(3)= P(X= 3)= 1/8
Histogram
0
1/8
2/8
3/8
4/8
0 1 2 3
x-values
P(x
)-va
lues
Line and Bar Graphs
Line Graph
0
1/8
2/8
3/8
4/8
0 1 2 4
x-values
p(x
)-va
lues
`
Bar Graph
0
1/8
2/8
3/8
4/8
0 1 2 4
x-values
p(x
)-va
lues
`
Expectation
Arithmetic Mean
Consider 10 hypothetical test scores: 65, 90, 70, 65, 70, 90, 80, 65, 90, 90
Calculate the mean as follows:
n
xxxx n
21
5.779010
480
10
170
10
265
10
3
10
904801702653
x
Expectation
We may express the arithmetic mean as:
As the number of repetition increases
kk xn
fx
n
fx
n
fx 2
21
1
iii pxn
f