• 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: x
1 x
2 x
3
. . .
x n
Probability: p
1 p
2 p
3
. . .
P n
__________________________________________________________________________
• The following must be true:
0
 p i

1 and i n 

1 p i

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?

• 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?

• Suppose that X is a discrete random variable whose distribution is
________________________________________________
Value of X: x
1 x
2 x
3
. . .
x n
Probability: p
1 p
2 p
3
. . .
P n
__________________________________________________________________________
•
To find the mean of X, multiply each possible value by its probability, then add all the products.
 x
 x
1 p
1
 x
2 p
2
 x
3 p
3

...
x n p n
 i n 

1 x i p i

•
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?

• 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?

• 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?

• 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?

• 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?