Topic 4 - Discrete Random Variables

Question 1

What is a random variable?

Answer 1

Given an experiment with sample space S, a random variable is a function from the sample space S to the real numbers R

So a random variable X assigns a numerical value X(s) to each possible outcome s of the experiment

For example the random variable where rolling a dice could be the ‘number of times 4 comes up’

Question 2

What is the difference between a discrete and continuous random variable?

Answer 2

See notes image

Question 3

Give 2 examples of a random experiment stating what the X and random variable would/could be

Answer 3

Random Experiment: Roll a dice twice
Let X be the r.v. ‘number of times 4 comes up’
then X can take value 0, 1, or 2

Random Experiment: Toss a coin twice
Sample space of four outcomes {HH, HT, TH, TT}
Let X be the r.v. ‘number of heads’, which can take values 0, 1 and 2.
X(HH)=2; X(HT)=X(TH)=1, X(TT)=0

Question 4

What is the random variable denoted as?

Answer 4

Uppercase X

Question 5

What is a probability distribution function (PDF)/probability mass function (PMF)?

Answer 5

A function which describes the probability that random variable X takes specific value x

Written as: P(X = x)

Question 6

What is a probability distribution function (PDF) also known as?

Answer 6

Probability mass function (PMF)

Question 7

What is a probability mass function (PMF) also known as?

Answer 7

Probability distribution function (PDF)

Question 8

What are the properties of probability distribution?

Answer 8

The probabilities can be greater than or equal to 0 but must be less than or equal to 1

The individual probabilities must sum to 1

Question 9

What is the difference between the probability mass/distribution function and the cumulative distribution function?

Answer 9

Probability mass/distribution function is the probability that X takes the value of x … P(X = x)

Cumulative distribution function is the probability that X takes a range of x values … P(X<=x0) for example

Question 10

What is the expected value or mean of a discrete distribution?

Answer 10

Expected value denoted by E(x)

E(x) = the sum of (the variable x multiplied by the probability of that x occurring)

Question 11

What must you remember about expected value/mean?

Answer 11

E(x) (the sum of the variable x multiplied by the probability of that x occurring) equals 1

Question 12

How do you calculate the variance of a discrete random variable X?

Answer 12

Variance (sigma^2) = E(X-mu)^2 = sum of (x - mu)^2 * P(x)

… variance (sigma squared) is equal to the sum of all the x values minus the mean squared times by the respective probabilities

… for each x value minus the mean mu squared, you times by the respective probability

Question 13

How do you calculate the standard deviation of a discrete random variable X?

Answer 13

Standard deviation (sigma) = the root of the variance = the root of E(X-mu)^2 = the root of the sum of (x - mu)^2 * P(x)

… standard deviation (sigma) is equal to the root of the sum of all the x values minus the mean squared times by the respective probabilities

… for each x value minus the mean mu squared, you times by the respective probability

Question 14

What is the variance if all the values take the form of a constant a and the expected value is also a?

Answer 14

Variance is 0 as all values are equal to constant a

Question 15

What is the expected value if all the values take the form of a constant a?

Answer 15

E (a) = a

Question 16

When you have a constant say b multiplied by the discrete random variable X what must you do?

Answer 16

E (bX) = bE (X)

When you have a constant say b multiplied by the discrete random variable X you multiply the same constant b by the expected value of the discrete random variable X

Question 17

How would you work out the variance when you have a constant say b multiplied by the discrete random variable X?

Answer 17

Var (bX) = b^2 * sigma^2 (for x)

NOTE that even though the constant is b that is being multiplied by the discrete random variable X, you still multiply the variance (sigma^2 for x) by b^2 (b squared) to find the variance when the discrete random variable X is multiplied by constant b

Question 18

What is key to remember about the variance of a constant?

Answer 18

It is 0

Question 19

What is key to remember about what to do when you have a constant being multiplied by the discrete random variable X?

Answer 19

Take the constant to the outside and multiply the expected value of the discrete random X by the constant
… E (bX) = bE (X)

Question 20

Applying what we have learned, what is the expected value of Y when it is equal to a + bX (… Y = a + bX)?

Answer 20

E (Y) = E (a + bX) = a + bE (X)

this means that the expected value of Y is equal to the expected value of a + bX which is equal to (applying the principles we have learned) a plus b multiplied by the expected value of the discrete random variable X

Question 21

Applying what we have learned, what is the variance of Y when it is equal to a + bX (… Y = a + bX)?

Answer 21

Sigma^2 for Y = Var (a + bX) = b^2 * sigma^2 (for x)

Variance (sigma squared) for Y is equal to the variance of a + bX which is equal to the just the variance of x multiplied by the constant squared as the constant a has a variance of 0 (as learned earlier)

Question 22

Applying what we have learned, what is the standard deviation of Y when it is equal to a + bX (… Y = a + bX)?

Answer 22

Sigma^2 for Y = Var (a + bX) = b^2 * sigma^2 (for x)

Sigma (for Y) = b * sigma (for x)

Variance (sigma squared) for Y is equal to the variance of a + bX which is equal to the just the variance of x multiplied by the constant squared as the constant a has a variance of 0 (as learned earlier)

Standard deviation of Y is equal to the constant b multiplied by sigma for x (the root of b squared times sigma squared)

Question 23

What is the variance of 2 random variables X and Y?

Answer 23

Var (X + Y) = Var (X) + Var (Y) + 2 cov

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