CHS Statistics - Chapter 4: Discrete Probability Distributions

random variable

an outcome of a probability experiment that is a count or measure

• x represents a value associated with each outcome of a probability experiment

• random indicates that x is determined by chance

2 types of random variables

discrete and continuous

discrete RV

has a finite/countable # of possible outcomes that can be listed

• represents counted data

• integers

• values are points on a number line

continuous RV

has an uncountable # of items

• represents measured data

• includes fractions/decimals

• represented by interval on a number line

discrete probability distribution

lists each possible value that random variable can assume, together with its probability

What 2 conditions must a discrete probability distribution satisfy?

1. The probability of each of value of the DRV is between 0 and 1, inclusive.

2. The sum of all the probabilities is 1.

What type of graph can a discrete prob. dist. be graphed with? Why?

a r.f. histogram b/c the probabilities represent r.f.s

how to construct a discrete prob. dist.

Let x be a DRV w/ possible outcomes x₁, x₂,…,x_n

1. Make a frequency distribution for the possible outcomes

   • Note: Decimals in table, fractions below

2. Find the sum of the frequencies

3. Find the probability of each possible outcome - divide its f by the sum of frequencies

4. Check that each prob. is beween 0 and 1, inclusive, and that the sum of probs. is 1

mean, variance, & standard deviation (discrete prob. dist.) on calculator

• x-values in L₁

• P(x)-values (FRACTIONS!!!) in L₂

• 1VarStats (FreqList = L₂)

expected value

E(x), same as mean

• Ex. For ticket costs, x = prize-cost of ticket, P(x) = prob. of winning prize

binomial experiment

has fixed # of independent trails, only 2 possible outcomes: success or failure

1. P(success) is the same for each trial

2. the random variable x counts the # of successes

3. x = 0, 1, 2, 3, 4, 5, …, n

binomial distribution symbols

• n = # of trials

• p = P(success)

• q = P(failure)

  • p + q = 1

• x = # of successes in n trials

creating a binomial prob. dist.

• x = 0, 1, 2, 3, …, n

• P(x) = binompdf of RV x

mean, variance, and standard deviation of binomial distributions

• mean = n • p

• variance = n • p • q

• standard deviation = sqrt(n • p • q)

binompdf vs. binomcdf

• binompdf = single probability

  • Ex. x = 4

• binomcdf = automatically sums P(0) + P(1) + P(2) + … + P(x)

  • x = where you tell it to stop (x-value)

geometric discrete prob. dist.

key phrase: first time

properties:

1. trials repeat until a success occurs

2. trials are independent of each other

3. P(success) is the same for each trial

4. the random variable x represents the number of the trial when the first success occurs

   • x = 1, 2, 3, …

   • p = P(success)

   • q = P(failure)

mean, variance, and standard deviation of a geometric dist.

• mean = 1/p

• variance = q/p²

• standard deviation = sqrt(q/p²)

poisson discrete prob. dist.

key phrase: gives mean/average

properties:

1. counts the number of times, x, an event occurs in a given interval

2. P(success) is the same for each interval

3. the # of occurrences in one interval is independent of other intervals

   • x = 0, 1, 2, …

   • p = P(success)

   • q = P(failure)

mean, variance, and standard deviation of a poisson dist.

• mean = mean

• variance = mean

• standard deviation = sqrt(mean)