Discrete Random Variables and Distribution Function

Question 1

Discrete Probability Disttribution

Answer 1

A probability distribution that depicts the occurence of discrete outcomes
Sample space is finite, each elementary event is assigned a particular probability, the event pace is the set of all subsets.
For example, we could ask the discrete probability of rolling a 3 on a fair sided dice, which would be 1/6. This would be shown on some sort of graph with depicts the discrete probability for 3 (numbers on x axis, probability on y axis etc).

Question 2

Continuous Distribution

Answer 2

The probability of every individual elementary event must equal 0 because there are infinitely many of them.

Question 3

Random Variable

Answer 3

A real-valued random variable X is a function X : sample space -> real number.
In other words, everything in a sample space gets associated to a real number. For example, students can get associated a grade.

Question 4

Using Random Variables in Probability

Answer 4

We can use this by doing:
P(X = v)
Where v is some real number.
So it could be a probability of a student getting v grade.

Question 5

Discrete Definition in Probability

Answer 5

Countable, finite.

Question 6

Example with only random variable

Answer 6

Lets say we have a sample space {H, T}
We could associated X(H) = 1 and X(T) = 0.
Therefore if we wanted P(X < 1), it would be 50% since we only have one other option, which is X being 0.

Question 7

Probability of a Random Variable

Answer 7

We can denote this as p_i, or:
P(X = v_i) = p_i
where v_i is some value.

Question 8

Σ p_i

Answer 8

1

Question 9

1 - P(X >= v)

Answer 9

P(X < v)

Question 10

Cumulative Distribution Function

Answer 10

The CDF of a random variable is denoted with F_X, with the main representation being:
F_X(v) = P(X <= v)
where v is some random variable.
This simply denotes the cumulative distribution for random variables smaller than v.

Question 11

Properties of F_X

Answer 11

• F_X(v_1) <= F_X(v_2) for any v_1 < v_2, so F_X(v) : Real Numbers -> {0,1} is monotonically increasing.
• The limit to negative infinity of F_X(v) is 0, and the limit to infinity for F_X(v) is 1, therefore F_X is strictly increasing from 0 to 1 over real numbers.
• For any v_1 < v_2, the difference between F_X(v_2) - F_X(v_1) equals to the probability of P(v_1 <= X < v_2) which exists in {0,1}

Question 12

Uniform Discrete Variable

Answer 12

Represented as X_n, where n has equally likely outcomes and value v_1 has the probability:
p_1 = 1/n, i = 1…,n.

Question 13

Bernoully Distribution

Answer 13

Bernoully variable X with parameter p is defined as P(X=1) = p = 1 = 1 - P(X=0) = 1-q, where p is either 0 or 1 and q = 1 - p.
Essentially, it represents pairs where X=0 and X=1.

Question 14

Binomial Distribution

Answer 14

Binomial variable X ~ B(n,p) is the number of successes in a sequence of n independent Bernoully distributions with parameters p which are either 0 or 1.