Topic 4 - Discrete Random Variables Flashcards
What is a random variable?
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’
What is the difference between a discrete and continuous random variable?
See notes image
Give 2 examples of a random experiment stating what the X and random variable would/could be
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
What is the random variable denoted as?
Uppercase X
What is a probability distribution function (PDF)/probability mass function (PMF)?
A function which describes the probability that random variable X takes specific value x
Written as: P(X = x)
What is a probability distribution function (PDF) also known as?
Probability mass function (PMF)
What is a probability mass function (PMF) also known as?
Probability distribution function (PDF)
What are the properties of probability distribution?
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
What is the difference between the probability mass/distribution function and the cumulative distribution function?
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
What is the expected value or mean of a discrete distribution?
Expected value denoted by E(x)
E(x) = the sum of (the variable x multiplied by the probability of that x occurring)
What must you remember about expected value/mean?
E(x) (the sum of the variable x multiplied by the probability of that x occurring) equals 1
How do you calculate the variance of a discrete random variable X?
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
How do you calculate the standard deviation of a discrete random variable X?
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
What is the variance if all the values take the form of a constant a and the expected value is also a?
Variance is 0 as all values are equal to constant a
What is the expected value if all the values take the form of a constant a?
E (a) = a
When you have a constant say b multiplied by the discrete random variable X what must you do?
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
How would you work out the variance when you have a constant say b multiplied by the discrete random variable X?
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
What is key to remember about the variance of a constant?
It is 0
What is key to remember about what to do when you have a constant being multiplied by the discrete random variable X?
Take the constant to the outside and multiply the expected value of the discrete random X by the constant
… E (bX) = bE (X)
Applying what we have learned, what is the expected value of Y when it is equal to a + bX (… Y = a + bX)?
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
Applying what we have learned, what is the variance of Y when it is equal to a + bX (… Y = a + bX)?
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)
Applying what we have learned, what is the standard deviation of Y when it is equal to a + bX (… Y = a + bX)?
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)
What is the variance of 2 random variables X and Y?
Var (X + Y) = Var (X) + Var (Y) + 2 cov
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