Comparing and Selecting Discrete Probability Distributions

When to use each distributions?

Uniform Distribution

Parameters

n : number of possible outcomes

Definition of Random Variable x

x maps a specific outcome to a probability

Range of values for x

Dependent on event. If its a dice, its 6 If its a 4 color spinner, its 4 Depends, depends

Probability Formula

P(X=x) = 1/n

Expectation Formula

E(x) = $\sum xP(x)$

Identifying Characteristics

All probabilities are the same its $\frac{1}{n}$

Binomial Distribution

Parameters

n : number of trials p : probability of success q : probability of failure

Definition of Random Variable x:

x is the number of specific successes

Range of Values for x:

0 to n

Probability Formula

P(X=x) = ${}_n{C}_x{p}^x{q}^{n-x}$

Expectation Formula

E(X) = np

Identifying Characteristics

• Bernoulli Trials
• Independent Trials

Geometric Distribution

Parameters

n : number of trials p : probability of success q : probability of failure Exact same as Binomial Distribution

Definition of Random Variable x:

Number of failures before a success

Range of Values for x:

0 → n-1 cause remember, it is before

Probability Formula

P(X=x) = $q^{x}p$ discrete### Expectation Formula E(x) = q/p

Identifying Characteristics

• Waiting time distribution. You are waiting until the success occurs
• Bernoulli Trials
• Independent Events

Keywords

• Waiting for success to occur at a certain spot. Eg, win first try, win 2nd try.
• Fewer than

Hyper Geometric Distribution

Parameters

n : population a : number of available successes r : number of selections

Definition of Random Variable X:

represents the number of successful selections

Range of Values for x

0 to r

Probability Formula

P(X=x) = $\frac{{}_a{C}_x{\ast }_{n-a}{C}_{r-x}}{{}_n{C}_r}$

Expectation Formula

E(x) = ra/n

Identifying Characteristics

• Dependent Trials
• Bernoulli Trials