Probability III

Question 1

What are two types of quantitative data?

Answer 1

Continuous and Discrete.

Question 2

Define Random Variables

Answer 2

Assumes a numerical value for each outcome of an experiment.

Question 3

Type of random variable that are whole numbers, obtained by counting and usually finite number of values.

Answer 3

Discrete Random Variable. Discrete Ex. Number of tails in 2 coin tosses. Random Variable X = 0, 1, 2.

Question 4

Type of random variable that are whole or fractional numbers, obtained by measuring, and be any infinite number of values in an interval.

Answer 4

Continuous Random Variable. Continuous Ex. Weight of a student (e.g., 115, 156.8, etc.)

Question 5

What are the 5 standards that make a discrete probability model?

Answer 5

1. List of all possible [x, p(x)] pairs, where x is the value of random variable (outcome) and p(x) is the probability associated with the value.
2. Mutually exclusive (No Overlap)
3. Collectively exhaustive (Nothing left out)
4. 0 <= p(x) <= 1.
5. Σp(x) = 1.

Question 6

Define the expected value of a probability distribution.

Answer 6

Mean of probability distribution.

Weighted average of all possible values.

µ = E(x) = Σ x • p(x).

Question 7

Define the variance of a probability distribution.

Answer 7

Weighted average squared deviation about the mean.

σ² = E[(x - µ)²] = Σ (x - µ)² • p(x).

Question 8

We usually slice up all the possible values into bins and then count the number of cases that fall into each bin. The bins, together with these counts, give the ____________ of the quantative variable and provide the building blocks for the display of the ___________, called a ________. By representing the counts as bars and plotting them against the bin values, the ________ displays the _________ at a glance.

Answer 8

distribution, distribution, histogram, histogram, distribution.

Question 9

Bernoulli Trials.

Answer 9

The Bernoulli Trials is a single performance of a well-defined experiment which Bernouli formatted to answer questions in a specific setup. It is used to build a wide variety of useful probability models.

The setup is

1. There are only two possible outcomes (called success and failure).
2. The probability of success, denoted p(x), is the same on every trial.
3. The trials are independent.

Note : A wide variety of probabilistic models, (Geometric, Binomial, Poisson, …… ), follow this setup; each one just answers a different question.

Question 10

This discrete probability distribution model uses Bernoulli trials to answer questions involving the interest in the number of successes (X) in a specified number of trials. What type of probability is this?

Answer 10

Binomial probability.

Question 11

It takes two parameters to define this _______ model: the number of trials, n and the probability of success, p. We denote this model _________.

Answer 11

Binomial, Binom(n,p).

Question 12

This model was derived to approximate the Binomial model when the probability of a success, p, is very small and the number of trials, n, is very large.

Answer 12

Poisson model.

Question 13

Describe a discrete probability distribution function.

Answer 13

1. Type of Model
   * Representation of Some Underlying phenomenon.
2. Mathematical Formula
3. Represents Discrete Random Variable.
4. Used to Get Probabilities for each X value.

Question 14

Binomial Distribution Characteristics

(These are only confined to the Binomial Probabilistic Model). T/F?

Mean = ?

Standard Deviation = ?

Answer 14

True.

µ = np.

σ = [np(1 - p)]^½

Question 15

Give the Binomial Probability Distribution Function.

Answer 15

P(X=x) = _nC_x •p^{<em>x </em>}•(1 - p)^<em>n-x</em>

where _nC_x = n! / x!(n-x)!

Question 16

Find the probability, P(X <= 5), of a binomial model s.t. P(x) is the probability of x and N = 6.

Answer 16

P(X <= 5) = P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4) + P(x=5) = 1 - P(x=6).

Question 17

Find the probability, P(X >= 2), of a binomial model s.t. P(x) is the probability of x and N = 6.

Answer 17

P(X >= 2) = P(x=2) + P(x=3) + P(x=4) + P(x=5) + P(x=6) = 1 - [P(x=0) + P(x=1)].

Question 18

Binomial Distribution Properties.

Answer 18

1. Sequence of n Identical Trials.
2. There are only two possible outcomes (called success and failure).
3. The probability of success, denoted p(x), is the same on every trial.
4. The trials are independent.