Transcript Random Variables

Random Variables
A numerical variable whose value
depends on the outcome of a chance
experiment is called a random
variable.
P(X = 4) is the probability of X having
the value of 4, where X is a random
variable
Continuous Random Variables



Probability distribution is
displayed in a density
curve, where the
probability equals the
proportion of area shaded
under the curve
Set of possible outcomes
consists of entire intervals
The probability of any
individual outcome is 0.
1.0
0.75
0.5
0.25
0.00
0
.25
.5
.75
1
P(.5 < x <.75) = .25
0.40
0.30
y
0.20
0.10
0.00
-3
-2
-1
normalDensit x   
y
P(x < -1) = .16
0
x
1
2
P(x = 1) = 0
Discrete Random Variables

Probability distribution is displayed in a table
and/or a probability (relative frequency)
histogram
 Set of possible outcomes consists of individual
values
X
Probability
x1
p1
x2 x3
p2 p3
0< pi < 1; Σ pi = 1
…
…
xk
pk
Example

Let X be a random variable for the number of
heads in 2 tosses of a coin. What is the
probability distribution for X? Draw the histogram
for the distribution.
X 0
1
P .25 .5
2
.25
0.6
0.5
0.4
0.3
0.2
0.1
0
01
2
1
23
Expected value (mean) of a
Discrete Random Variable


The idealized average or the long run average value
A weighted average, because the single probability of
each outcome is not equal
E ( X )   X   xi pi

What is the expected value for the number of heads in 2
tosses of a coin?
 0(.25) + 1(.5) + 2(.25) = 1
 We expect to get an average of 1 head in 2 tosses of
a coin
Variance & Standard deviation of a
Discrete Random Variable
 Variance,σ2x
, is the average squared
deviation of the values of the variable from
their mean
2
2
Var ( X )   x   ( xi   x ) pi
deviation, σx , and variance
measure the variability of the distribution
about the mean StDev( X )     2
 Standard
x
x

What is the standard deviation for the number of
heads in 2 tosses of a coin?
X
0
1
2
P
.25
.5
.25
Var(x) = .25(0 - 1)2 + .5(1 - 1)2 + .25(2 - 1)2 = .5
 StDev(x) = .5 = .707

Mean & Variance for Continuous
Random Variables
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The mean:



The standard deviation & variance:

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The balancing point of graph
For normal distributions,  X is the value at the center
Will be given to you
For normal distributions, use z-scores to find
probabilities
Examples
–
Determine the expected value and standard deviation for the
probability distribution defined below:
X
•
1
2
3
4
P(X) .3
.2
.4
.1
A club sells raffle tickets for $5 each. There are 10 prizes of $25
and one prize of $100. If 200 tickets are sold, and you bought
one of them, what are your expected winnings? How much, on
average, is the club profiting per ticket?
•
Insurance companies compute expected values so
that they can set their rates at profitable but
competitive levels. A 64-year-old man obtains a
$10,000 one-year life insurance policy at a cost of
$600. Based on past mortality experience, the
insurance company estimates that there is at 0.963
chance that this man will live for at least one year.
How much can the insurance company expect to earn
on this policy?
•
The Wisconsin Lottery recently
had a scratch-off game called
“Big Cat Cash.” It cost $1.00 to
play. The probabilities of
winning various amounts are
listed.
–
Find and interpret the expected
amount of winnings when
playing one game.
–
Calculate the standard
deviation.
Winnings
$1
$2
$3
$18
$50
$150
$900
$0
Probability
1/10
1/14
1/24
1/200
1/389
1/20,000
1/120,000
• The XYZ Office Supplies Company sells calculators in bulk at
wholesale prices, as well as individually at retail prices. Next year’s
sales depend on market conditions, but executives use probability
to find estimates of sales for the coming year. The following tables
are estimates for next year’s sales. What profit does XYZ Office
Supplies Company expect to make for the next year if the profit
from each calculator sold is $20 at wholesale and $30 at retail?
– Wholesale Sales # sold
2000
5000
10000
20000
0.1
0.3
0.4
0.2
# sold
600
1000
1500
P(x)
0.4
0.5
0.1
P(x)
– Retail Sales