Chapter 7 - Random Variables

Front 1

Probability Distribution (of a discrete random variable)

Back 1

A list of probabilities associated with each of its possible values. The probabilities must satisfy the two requirements: Every probability is between 0 and 1 and the sum of the probabilities must equal 1.

Hint 1

Front 2

Discrete Random Variable

Back 2

A random variable, X, has a countable number of possible values.

Hint 2

Front 3

Random Variable

Back 3

A variable whose value is a numerical outcome of a random phenomenon

Hint 3

Example:
When tossing a coin and if X is the number of heads, than X is a random variable because its values vary when the coin tossing is repeated.

Front 4

Continuous Random Variable

Back 4

A variable that takes all values in an interval of numbers. Its probability distribution can be described with a density curve. Probability for a certain interval can be found by taking the area of the space between two values in its density curve. The probability of an individual outcome is 0.

Hint 4

Example:
When observing the amount of rain falling in a city and X is rain in inches, X is a continuous random variable because there are infinite numbers of values.

Front 5

Normal Distribution

Back 5

One type of continuous distribution. The normal distribution has a mean of 0 and standard deviation of 1.

Hint 5

The bell curve histogram.

Front 6

Mean of Any Discrete Random Variable

Back 6

It is an average of the possible outcomes but a weighted average in which each outcome is weighted by its probability.

Hint 6

Payoff x: $0 $500
Probability: 0.999 0.001

$500(0.001) + $0(,999) = $0.50.

Front 7

Variance

Back 7

Hint 7

For the set 4, 2, 5, 8, 6, we calculate the mean to be 5.
Then the sum of (x-mean)^2=20
so variance=20/5=4

Front 8

Standard Deviation

Back 8

√Variance

Hint 8

See Variance.
If variance is 4, standard deviation is √4=2

Front 9

Standardized Variable (Z-Score)

Back 9

Hint 9

Example calculation.

Front 10

Sampling Distribution

Back 10

The probability distribution of random variables

Hint 10

Front 11

Law of Large Numbers

Back 11

In the long run, the proportion of outcomes taking any value gets close to the probability of the value. The average outcome gets close to the distribution mean.

Hint 11

Front 12

Law of Small Numbers

Back 12

The incorrect assumption that short sequences of random events will show the kind of average behavior that occurs only in the long run.

Hint 12

RandInt(0,100,2)
this will not give us an accurate average of what will happen in the long run

Front 13

Rule for Means

Back 13

Rule 1:

Hint 13

Rule 2:

Front 14

Rule for Variances

Back 14

Rule 1:

Hint 14

Rule 2: