Discrete Random Variables

Question 1

what does the random variable X represent?

Answer 1

a single trial/ experiment
* consisst of discrete outcomes each with a distinct probability

Question 2

what does P(X= x) mean?

Answer 2

the probability that the outcome of the random variable X is the specific outcome x, eg: 1,2,3,4,5,6

Question 3

what is the expected value E(X) ?

Answer 3

the mean outcome we would expect if we were to do our experiment many times

Question 4

How would you find the expected value of the discrete random variable X?

E(X)

Answer 4

E(X)= Σ [(x ) (P(X=x)]

Question 5

Find the expected value E(x) from the following tables

Answer 5

Question 6

what do you notice about the expected value of a symetrical distribution?
eg:

Answer 6

the expected value is the central value

Question 7

Solve the following equation

Answer 7

Question 8

If x is the original random variable, what would 2x and x squared represent?

Answer 8

Question 9

how would you find E(X^2)?

Answer 9

Σ [x^2][P(X=x)

Question 10

Find E (X^2)

Answer 10

Question 11

what is the formula for the Var(X) of a discrete random variable?

Answer 11

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

Question 12

Solve the following question

Answer 12

Question 13

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

Answer 13

1. aE(X) + b
2. cE(X)
3. d

Question 14

what are the rules for variance of a linear function?
1. Var (aX)
2. Var (aX+b)
3. Var (c)

Answer 14

1. a^2 Var(X)
2. a^2 Var(X)
3. 0
   * no variation in a constant
   * adding a constant does not change the variation as all values will have this added to

Question 15

simplify the following

Answer 15

Question 16

what are the simplified results for
1. E (X1 + X2)
2. E (X1- X2)

Answer 16

1. E(X1) + E(X2)
2. E(X1) - E(X2)

Question 17

What are the simplified results for
* Var (X1 + X2)
* Var (X1- X2)

Answer 17

• Var (X1) + Var (X2)
• Var (X1) + Var (X2)

Question 18

what do these linear combinations of random variables become?
1. E( aX + bY)
2. Var (aX + bY)

Answer 18

1. aE(X) + bE(Y)
2. a^2 Var(X) + b^2 Var(X)

Question 19

What is the variance?

Answer 19

a measure of how data points differ from the mean

Question 20

what is standard deviation?

Answer 20

square root of Var(X)

Question 21

Use calculator to find the following

Answer 21