Defining probabilities: random variables
 Examples:
 Out of 100 heart catheterization procedures performed at a local
hospital each year, the probability that more than five of them will
result in complications is
__________
 Drywall anchors are sold in packs of 50 at the local hardware store.
The probability that no more than 3 will be defective is
__________
 In general, ___________
1
EGR 252 - Ch. 3
Discrete random variables
 Example:
 Look back at problem 2.53, page 55. Assume someone spends
$75 to buy 3 envelopes. The sample space describing the
presence of $10 bills (T) vs bills that are not $10 (N) is:
_____________________________
 The random variable associated with this situation, X, reflects
the outcome of the choice and can take on the values:
_____________________________
2
EGR 252 - Ch. 3
1
Discrete probability distributions
 The probability that there are no $10 in the group is
P(X = 0) = ___________________
The probability distribution associated with the number of $10
bills is given by:
x
0
1
2
3
P(X = x)
3
EGR 252 - Ch. 3
Another example
 Example 3.8, pg 80
P(X = 0) =
_____________________
4
EGR 252 - Ch. 3
2
Discrete probability distributions
 The discrete probability distribution function (pdf)
 f(x) = P(X = x) ≥ 0
 Σx f(x) = 1
 The cumulative distribution, F(x)
 F(x) = P(X ≤ x) = Σt ≤ x f(t)
5
EGR 252 - Ch. 3
Probability distributions
 From our example, the probability that no more than 2 of the
envelopes contain $10 bills is
P(X ≤ 2) = F(2) = _________________
 The probability that no fewer than 2 envelopes contain $10
bills is
P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - F(1) = ________________
6
EGR 252 - Ch. 3
3
Another view
 The probability histogram
0.45
0.4
0.35
f(x)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
x
7
EGR 252 - Ch. 3
Your turn …
 The output from of the same type of circuit board from two assembly lines is mixed
into one storage tray. In a tray of 10 circuit boards, 6 are from line A and 4 from
line B. If the inspector chooses 2 boards from the tray, show the probability
distribution function associated with the selected boards being from line A.
x
P(x)
0
1
2
8
EGR 252 - Ch. 3
4
Continuous probability distributions
 Examples:
 The probability that the average daily temperature in Georgia
during the month of August falls between 90 and 95 degrees is
__________
 The probability that a given part will fail before 1000 hours of use is
__________
 In general, __________
9
EGR 252 - Ch. 3
Understanding continuous distributions
 The probability that the average
daily temperature in Georgia
during the month of August falls
between 90 and 95 degrees is
-5
-3
-1
1
3
5
 The probability that a given part
will fail before 1000 hours of
use is
0
10
5
10
15
20
25
30
EGR 252 - Ch. 3
5
Continuous probability distributions
 The continuous probability density function (pdf)
f(x) ≥ 0, for all x ∈ R
f ( x )dx 1
b
P(a X
b)
f ( x )dx
a
 The cumulative distribution, F(x)
x
F ( x ) P(X
x)
f (t )dt
11
EGR 252 - Ch. 3
Probability distributions

Example: Problem 3.7, pg. 88
f(x) =
{
x,
2-x,
0,
0<x<1
1≤x<2
elsewhere
1st – what does the function look like?
a) P(X < 120) = ___________________
b) P(50 < X < 100) = ___________________
12
EGR 252 - Ch. 3
6
Your turn
 Problem 3.14, pg. 89
13
EGR 252 - Ch. 3
Joint probability distributions
 The probabilities associated with two things both happening,
e.g. …
 probability associated with the hardness and tensile strength of an
alloy
 f(h, t) = P(H = h,T = t)
 probability associated with the lives of 2 different components in an
electronic circuit
 f(a, b) = P(A = a, B = b)
 probability associated with the diameter of a mold and the diameter
of the part made by that mold
 f(d1, d2) = P(D1 = d1, D2 = d2)
14
EGR 252 - Ch. 3
7
Example
 Statistics were collected on 100 people selected at random who
drove home after a college football game, with the following
results:
Accident
No
accident
TOTAL
Drinking
7
23
30
Not
drinking
6
64
70
TOTAL
13
87
100
15
EGR 252 - Ch. 3
Joint probability distribution
 The joint probability distribution or probability mass function for this
data:
Accident
No
accident
g(x)
Drinking
0.07
0.23
0.3
Not
drinking
0.06
0.64
0.70
h(y)
0.13
0.87
1
 The marginal distributions associated with drinking and having an
accident are calculated as:
_______________________________________
16
EGR 252 - Ch. 3
8
Conditional probability distributions
 The conditional probability distribution of the random variable Y
given that X = x is
f ( y |x )
f ( x, y )
g( x )
 For our example, the probability that a driver who has been
drinking will have an accident is
f(accident | drinking) = ___________________
(note: refer to page 95 of your textbook for the conditional probability that a variable falls
within a range.)
17
EGR 252 - Ch. 3
Statistical independence
 If f(x | y) doesn’t depend on y, then f(x | y) = g(x) and
f(x, y) = g(x) h(y)
 X and Y are statistically independent if and only if this holds
true for all (x, y) within their range.
 For our example, are drinking and being in an accident
statistically independent? Why or why not?
18
EGR 252 - Ch. 3
9
Continuous random variables
 f(x, y) is the joint density function of the continuous
variables X andY, and the associated probability is
P[( x, y ) A
f ( x, y )dxdy
A
for any region A in the xy plane.
 The marginal distributions of X andY alone are
g( x )
f ( x, y )dy and h( y )
f ( x, y )dx
 See examples 3.15, 3.17, 3.19, 3.20, and 3.22 (starting on
pg. 93)
19
EGR 252 - Ch. 3
10