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Random Variables
When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctorâs office, their height, weight, temperature and blood pressure are recorded. These observations vary from patient to patient, hence they are called variables. We tend to call these random variables, because we cannot predict what their value will be for the next trial of the experiment (for the next patient).
Rather than repeat and write the words height, weight and blood pressure many times, we tend to give random variables names such as X, Y.. .. We usually use capital letters to denote the name of the variable and lowercase letters to denote the values.
Random Variables
A Random Variable is a rule that assigns a number to each outcome of an experiment.
Example: An experiment consists of rolling a pair of dice, one red and one green, and observing the pair of numbers on the uppermost faces (red first). We let X denote the sum of the numbers on the uppermost faces. Below, we show the outcomes on the left and the values of X associated to some of the outcomes on the right:
{ (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }
Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 .. .
Dice example
(c) We could also define other variables associated to this experiment. Let Y be the product of the numbers on the uppermost faces. What are the values of Y associated to the various outcomes?
Dice example
- (1, 1) Outcome Y
- (2, 1), (1, 2)
- (3, 1), (1, 3)
- (4, 1), (1, 4)
- (5, 1), (1, 5)
- (6, 1), (1, 6) - (2, 2) Outcome Y - (3, 2), (2, 3) - (4, 2), (2, 4) - (5, 2), (2, 5) - (6, 2), (2, 6)
- (4, 3), (3, 4)
- (5, 3), (3, 5)
- (6, 3), (3, 6) - (4, 4) Outcome Y - (4, 5), (5, 4) - (4, 6), (6, 4) - (5, 5) - (5, 6), (6, 5) - (6, 6)
Coin example
Example: An experiment consists of flipping a coin 4 times and observing the sequence of heads and tails. The random variable X is the number of heads in the observed sequence. Draw a table that shows the possible values of X and the number of outcomes associated to each value.
Value of X outcomes no. of outcomes 4 HHHH 1 3 HHHT, HHTH, HTHH, THHH 4 2 HHTT, HTHT, HTTH, THHT, THTH, TTHH 6 1 HTTT, THTT, TTHT, TTTH 4 0 TTTT 1
Discrete vs. continuous random variables
For some random variables, the possible values of the variable can be listed in either a finite or an infinite list. These variables are called discrete random variables. Some examples:
Experiment Random Variable, X Roll a pair of six-sided dice Sum of the numbers Roll a pair of six-sided dice Product of the numbers Toss a coin 10 times Number of tails Choose a small pack of M&Mâs at random The number of blue M&Mâs in the pack Choose a year at random The number of people who ran the Boston Marathon in that year On the other hand, a continuous random variable can assume any value in some interval. Some examples:
Experiment Random Variable, X Choose a patient at random Patientâs Height Choose an apple at random at your local grocery store Weight of the apple Choose a customer at random at Subway The length of time the customer waits to be served
Dice example
Example If I roll a pair of fair six sided dice and observe the pair of numbers on the uppermost face, all outcomes are equally likely, each with a probability of 361. Let X denote the sum of the pair of numbers observed. We saw that a value of 3 for X is associated to two outcomes in our sample space: (2, 1) and (1, 2). Therefore the probability that X takes the value 3 or P(X = 3) is the sum of the probabilities of the two outcomes (2, 1) and (1, 2) which is 2
- That is P(X = 3) =
If X is a discrete random variable with finitely many possible values, we can display the probability distribution of X in a table where the possible values of X are listed alongside their probabilities.
Dice example
I roll a pair of fair six sided dice and observe the pair of numbers on the uppermost face. Let X denote the sum of the pair of numbers observed. Complete the table showing the probability distribution of X below:
{(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}
X P(X) 2 3 4 5 6 7 8 9
10 11 12
Probability Distributions
If a discrete random variable has possible values x 1 , x 2 , x 3 ,... , xk, then a probability distribution P(X) is a rule that assigns a probability P(xi) to each value xi. More specifically, I (^0) ⤠P(xi) ⤠1 for each xi. I (^) P(x 1 ) + P(x 2 ) + ¡ ¡ ¡ + P(xk) = 1.
Example An experiment consists of flipping a coin 4 times and observing the sequence of heads and tails. The random variable X is the number of heads in the observed sequence. Fill in probabilities for each possible values of X in the table below. X 0 1 2 3 4 P(X)?????
Coin Example
X 0 1 2 3 4
P(X)
Bar graphs of distributions
Example: An experiment consists of flipping a coin 4 times and observing the sequence of heads and tails. The random variable X is the number of heads in the observed sequence. The following is a graphical representation of the probability distribution of X.
0 1 2 3 4
Bar graphs of distributions
Example: The following is a probability distribution histogram for a random variable X.
1 2 3 4 5 6 X
1
2
What is P(X 6 5)?
Some gambling examples
Example: In a carnival game a player flips a coin twice. The player pays $1 to play. The player then receives $1 for every head observed and pays $1, to the game attendant, for every tail observed. Find the probability distribution for the random variable X = the playerâs (net) earnings.
There are 4 possible outcomes HH, HT , T H, T T. The return to the player in the case HH is 1, the return to the player in the case HT or T H is â1, and the return to the player in the case T T is â3. Hence P(X = 1) =
P(X = â1) =
and P(X = â3) =
Some gambling examples
A roulette wheel has 18 red numbers, 18 black numbers and 2 green numbers. When the wheel is spun and a ball dropped onto it, the ball is equally likely to land on any of the 38 numbers. When you bet $1 on red, I (^) if the ball lands on a red number you get your $1 back plus $1 profit, and I (^) if the ball lands on a black or a green number, you lose your initial dollar. What is the probability distribution for your earnings for this game if you bet $1 on red? There are only two outcomes: you win $1 or you get â$1.
P(X = 1) =
P(X = â1) =
Since 1838 is less than 1/2, and 2038 is greater than 1/2, you lose more often than you win at Roulette (naturally; otherwise the casino wouldnât offer it!)
