Discrete Random Variables Quiz

Study Notes

A variable whose outcome depends on a random experiment
A discrete RV can take on a finite or countably infinite number of values.
The probability distribution function (PDF) or probability mass function (PMF) is a table or function that lists all possible outcomes and their associated probabilities.
The notation for the probability of a specific value of a variable (e.g., X = x) is:
- P(X = x)
- P(x)
- p(x)
- f(x)
All these notations mean the same thing: the probability that the random variable X takes on the value x.

The sum of all probabilities for all possible values of the random variable must equal 1.
The probability of each individual outcome must be between 0 and 1 (inclusive).

Example: Let X be the amount of money a student earns per hour.
The possible values of X are 100, 120, 150, and 170.
The probabilities associated with each value are:
- P(100) = 0.15
- P(120) = 0.25
- P(150) = 0.45
- P(170) = 0.15
Note that the sum of all probabilities is 1.

Sometimes, a PMF is given with an unknown constant, "k".
To find this constant, we can use the fact that the sum of all probabilities must equal 1.
Example: find the value of k for the following PMF:
- P(y) = k for y = -4, 0, 3, 2, 8
  - To find k, we set the sum of the probabilities to 1 and solve for k:
    - k + 0.125 + 0.125 + 0.375 + 0.125 = 1
    - k = 0.25

Example: Find the probability that X is less than 4 for the following PMF:
- P(X = x) = x^2/55 for x = 1, 2, 3, 4, 5
- To calculate this, add the probabilities for all values of X less than 4.
- P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3)
- P(X < 4) = (1/55) + (4/55) + (9/55) = 14/55