Transcript 8-7 Cont`d Random Variables Notes

Random Variables
Definitions
• A random variable is a variable whose value is a numerical outcome of a
random phenomenon,.
• A discrete random variable X has a countable number of possible values.
The probability distribution of a discrete random variable X lists the
values of their probabilities:
________________________________________________
Value of X:
x1
x2
x3
...
xn
Probability:
p1
p2
p3
...
Pn
__________________________________________________________________________
• The following must be true:
n
0  pi  1
and
p
i 1
i
1
Example 1
• NC State posts the grade distributions for its courses
online. Students is Statistics 101 in the fall 2003 semester
earned 21% A’s, 43% B’s, 30% C’s, and 5% D’s. Choose
a students at random. What is the probability that the
student failed the course? What is the probability that the
student got a B or better?
Example 2
• Spell-checking software catches “non-word errors,” which result in a
string of letters that is not a word, as when “the” is typed as “teh.”
When undergraduates are asked to write a 250-word essay (without
spell-checking), the number X of non-word errors has the following
distribution:
_______________________________________
X:
0
1
2
3
4
P(x):
0.1
0.2
0.3
0.3
0.1
____________________________________________________________
• Write the event “at least one non-word error” in terms of X. What is
the probability of this event?
• Describe the event X ≤ 2 in words. What is its probability? What is
the probability that X < 2?
Mean of a Discrete Random Variable
• Suppose that X is a discrete random variable whose distribution is
________________________________________________
Value of X:
x1
x2
x3
...
xn
Probability:
p1
p2
p3
...
Pn
__________________________________________________________________________
• To find the mean of X, multiply each possible value by its probability,
then add all the products.
 x  x1 p 1  x 2 p 2  x 3 p 3  ... x n p n
n

x
i 1
i
pi
Example 1 Revisited
• NC State posts the grade distributions for its courses
online. Students is Statistics 101 in the fall 2003 semester
earned 21% A’s, 43% B’s, 30% C’s, 5% D’s and 1% F’s.
Choose a students at random. What is the mean grade for
this course?
Example 2 Revisited
• Spell-checking software catches “non-word errors,” which result in a
string of letters that is not a word, as when “the” is typed as “teh.”
When undergraduates are asked to write a 250-word essay (without
spell-checking), the number X of non-word errors has the following
distribution:
_______________________________________
X:
0
1
2
3
4
P(x):
0.1
0.2
0.3
0.3
0.1
____________________________________________________________
• What is the expected number of non-word errors in the 250-word
essay?
Example 3
• A single male driver aged 23 living in a suburb of a Midwestern city
pays a car insurance premium of $496 per year. The insurance
company’s actuarial model for this driver predicts the following
probabilities for 1 year: 0.065 for an accident averaging $4600 in
damages; 0.03 for $2400 in damages; and 0.015 for $1000 in damages.
What is the expected damage cost the insurance company should be
prepared to pay for such a driver? What amount is the policy expected
to contribute to the operation and profit of the company?
Example 4
• Two coins are tossed. If both land heads up, the player A wins $4
from Player B. If exactly one coin lands heads up, then Player b wins
$1 from Player A. If both land tails up, then Player B wins $2 from
Player A. Is this a fair game?
Example 5
• You pick a number from 1 – 6 and roll 3 dice. If the number you pick
comes up 3 times, you win $3. If it comes up twice, you win $2, and
once you win $1, otherwise you lose $1. What is your expected gain
or loss?