Problem Solving with DRVs
How do I find DRVs from worded contexts?
- Introduce a capital letter, , to represent numerical values
- Make sure you know exactly what represents
- List all the values can take
- Calculate the probability for each value of
- Put this information into a table
-
... ...
-
- If asked to comment on results
- Find to talk about the average result
- Find to talk about the spread of results
What if they introduce a new independent variable?
- Sometimes questions have two independent DRVs, and
- e.g. may be a coin and may be a 3-sided spinner
- Construct their tables
-
2 4 0.25 0.75 -
0 2 4 0.1 0.2 0.7
-
- To find
- List possibilities
- or
- Multiply probabilities then add
- 0.25 × 0.2 + 0.75 × 0.1
- List possibilities
- To find
- Either construct a full table for all possible values that can take
- There may be a lot!
- Then work out expectation from the table
- Or use the formula
- Find and from their tables above
- Either construct a full table for all possible values that can take
- Note that, if and are independent
- Then
- The expectation of a product is the product of the expectation
- These ideas can be applied when repeatedly using the same distribution,
- For example, flipping a coin, then flipping it again
- Create independent random variables
What if they introduce a new dependent variable?
- Sometimes questions introduce a new variable, , which depends on an old variable,
-
-3 1 2 3 0.25 0.25 0.25 0.25 - Let
- List values of
- They have the same probabilities as (as depends on )
-
0 8 5 0 0.25 0.25 0.25 0.25 -
0 5 8 0.5 0.25 0.25
-
- You can use
- calculate and from their tables
- You cannot use if depends on
- To find draw the table for all possible values of
-
-3 × 0 1 × 8 2 × 5 3 × 0 0.25 0.25 0.25 0.25 - Probabilities are the same as (as depends on )
- Then calculate expectation from this table
-
- Alternatively, substitute in
- So
- Find from the table of
- Find from the table of (by cubing the values)
- To find draw the table for all possible values of
What if there is conditional probability?
- Learn the conditional probability formula
- For example, for integer values, the probability that given that
Exam Tip
- Always draw out tables of values for each DRV - they really help in the exam!
Worked example
In a game, you can earn 2, 5 or 6 points. There is a 50% chance of earning 6 points and an equal chance of earning either 2 or 5 points.
The amount of money (£) won in a game is found by multiplying the number of points, , by the variable , where if or if .