Discrete Random Variables Flashcards
what does the random variable X represent?
a single trial/ experiment
* consisst of discrete outcomes each with a distinct probability
what does P(X= x) mean?
the probability that the outcome of the random variable X is the specific outcome x, eg: 1,2,3,4,5,6
what is the expected value E(X) ?
the mean outcome we would expect if we were to do our experiment many times
How would you find the expected value of the discrete random variable X?
E(X)
E(X)= Σ [(x ) (P(X=x)]
Find the expected value E(x) from the following tables
what do you notice about the expected value of a symetrical distribution?
eg:
the expected value is the central value
Solve the following equation
If x is the original random variable, what would 2x and x squared represent?
how would you find E(X^2)?
Σ [x^2][P(X=x)
Find E (X^2)
what is the formula for the Var(X) of a discrete random variable?
Var(X)= E(X^2) - E(X)^2
1. work out E(X)
2. find E(X^2) by using Σ (x^2)(P=x)
3. minus this from the E(X) squared
Solve the following question
what is the expectation for a linear function of a random variable?
1. E (aX +b)=
2. E (cX)=
3. E (d)=
rules of expectation
- aE(X) + b
- cE(X)
- d
what are the rules for variance of a linear function?
1. Var (aX)
2. Var (aX+b)
3. Var (c)
- a^2 Var(X)
- a^2 Var(X)
- 0
* no variation in a constant
* adding a constant does not change the variation as all values will have this added to
simplify the following
what are the simplified results for
1. E (X1 + X2)
2. E (X1- X2)
- E(X1) + E(X2)
- E(X1) - E(X2)
What are the simplified results for
* Var (X1 + X2)
* Var (X1- X2)
- Var (X1) + Var (X2)
- Var (X1) + Var (X2)
what do these linear combinations of random variables become?
1. E( aX + bY)
2. Var (aX + bY)
- aE(X) + bE(Y)
- a^2 Var(X) + b^2 Var(X)
What is the variance?
a measure of how data points differ from the mean
what is standard deviation?
square root of Var(X)
Use calculator to find the following
Prove the E(aX +b) = aE(X) +b
