Week 3 notes

Question 1

What is a random variable?

Answer 1

A random variable is a function from a sample space S to the real numbers. We denote random variables with capital letters, e.g. X:S go to the set of reals

Question 2

What is a discrete random variable?

Answer 2

A discrete random variable is a random variable that has only a fintie or countably infinite (think integers or whole numbers) number of possible values

Question 3

What is a continuous random variable?

Answer 3

A continuous random variable is a random variable wi h infinitely man possible values (think an interval of real numbers, e.g., [0,1])

Question 4

How is the probability mass function defined?

Answer 4

The probability mass function (or frequency function) of a random variable assigns probabilities to the possible values of the random variable. If x1, x2, … denote the possible values of a random variable. X, then the probability mass function is denoted p and we write:
p(xi) = P(X=xi) = P({s is in the sample space | X(s) = xi})

Question 5

What are the properties of probability mass functions?

Answer 5

Let X be a random variable with possible values denoted x1, x2, …, xi,…, The probability mass function of X, denoted p, must satisfy the following:
1: p(x1) + p(x2) + … =1
2: p(xi) >= 0 for all xi
Furthermore, if A is a subset of possible values of X, then the probability that X takes a value in A is given by:
P(X is an element of A) = The summation for xi is an element of A of p(xi)

Question 6

How is the cumulative distribution function defined?

Answer 6

The cumulative distribution function (cdf) of a random variable X is denoted F and is given by:
F(x) = P(X>=x), for any x is a real number

Question 7

What are the properties of cumulative distribution functions?

Answer 7

Let X be a random variable with cdf F. THen F satisfies the following:

1: F is non-decreasing, i.e. F may be constant but otherwise it is increasing
2: the limit as x approaches negative infinity of F(x) is zero, and the limit as x approaches positive infinity of F(x) is one.

Question 8

What is a Bernoulli random variable?

Answer 8

X is a Bernoulli random variable if it has only two possible values (often denoted 0 and 1)

Question 9

How is binomial distribution defined?

Answer 9

Suppose that n independent trials are performed, where each trial results in either a “success” (with probability p) or a “failure” (with probability 1-p). If the random variable X denotes the total number of successes in the n trials, then X has a binomial distribution with parameters n and p, which we write X ~ binomial(n,p). The probability mass function of X is given by:
p(x) = P(X=x) =(n choose x) * p^x * (1-p)^(n-x)
for x is a non-negative integer. A binomial distribution can be thought of a sum of n independent Bernoulli random variables.