Module 1 Notes

We often use the letter "p" to represent a probability. When writing specific probability statements, however, we will use _____ instead.

"Pr(Event)"

To describe a random variable completely, we need to know two things: _____

(1) every possible numeric outcome and (2) the probability of each outcome.

We can learn about probabilities in different ways. Some events have theoretical probabilities. When flipping a coin, for example, we know that the probability that the coin will land heads (or tails) up is ____

.5

A probability of ____ means that the event never happens—it is impossible

0 or 0%

Another key measure is the expected value of (X - E(X))2 — this is called the ____

variance of X.

The standard deviation helps us determine ____

which outcomes are more or less likely.

Remember that p_i is the probability that X takes the value ____

x_i.

The expected value or mean of random variable X is denoted E(X) or u; for a discrete random variable, the formula is _____

μ=E(X)=∑_i▒x_i p_i

The variance, written either as Var(X) or , is defined by the formula _____

σ^2=Var(X)=∑_i▒((x_i-E(X))^2∙p_i )

For the random variable in the preceding example, the standard deviation, denoted by the Greek letter, σ is _____

√1.40

It assigns the numerical value of ____ to an event occurring, the numerical value of _____ to the event not occurring

1 0

A probability of _____ means that the event always happens—it is certain.

1 or 100%

There are other rules for calculating means and variances when scaling or translating random variables that can help you save time. These rules will be demonstrated in the associated video. The rules are summarized below. 1). _____ 2). _____

1). If X is a random variable with mean E(X) and variance Var(X), then for any constant c, cX is a (new) random variable with mean cE(X) and variance c^2Var(X). 2). If X is a random variable with mean E(X) and variance Var(X), then for any constant d, d + X is a (new) random variable with mean d + E(X) and variance Var(X).

When rolling dice with six faces, we know that the probability that a die will land with any particular face up—either 1, 2, 3, 4, 5, or 6 — is _____

1/6 = .1667.

Because a 52-card deck includes 4 aces, we know the probability that a randomly selected card is an ace is ____

4/52 = .0769.

The Bernoulli distribution, including its mean and variance, will be important in discussing our second discrete distribution...the _____

Binomial.

The simplest discrete probability distribution is the ____

Bernoulli distribution.

Such an event is often called a _____

Bernoulli trial.

Example: Electric Motors. You sell large electric motors to a single customer. Based on your historical data, you know that demand for your motors from your main customer can be 0, 1, 2, or 4 (4 come on a pallet). The distribution is Demand (xi) 0 1 2 4 Probability (pi).40 .40 .10 .10 What is the expected demand for a week?

E(X) = (0)(.40)+(1)(.40)+(2)(.10)+(4)(.10) = 1.00

To compute the variance of demand for the motor example discussed, follow these steps: Step 1: _____

Determine E(X). From a previous calculation, we know it is 1.00.

To compute the variance of demand for the motor example discussed, follow these steps: Step 2: _____

Step 2. List all outcomes for (x_i-E(X))^2, their associated probabilities, and the products.

To compute the variance of demand for the motor example discussed, follow these steps: Step 3: _____

Step 3: Sum the products

In ______, 99.99966% of the products are expected to be free of defects or errors.

a six-sigma (±6σ) process

We learn the probability of other events empirically, using the ____

actual frequency that the event occurs.

We say that E(X) or μ is a measure of ____

central tendency for the random variable X.

A general rule of thumb for practical statistical applications is that about 95% of observed outcomes will ____

come within two standard deviations (±2σ) of the mean

An example of a _____ variable is the time it takes to process a customer order

continuous random

A _____ is one for which the outcomes are so numerous that they cannot be listed

continuous random variable

It is important to note that using only a sample of US families or new product introductions is not enough to _____, though we will talk about how to use samples of data in Module 3.

determine the true probability

There are two general types of random variables: ____ and _____

discrete and continuous.

A _____ is one for which all possible outcomes can be listed

discrete random variable

Var(X) or σ^2 measures the ______

dispersion of a random variable

The formula looks complicated, but a few simple examples will clarify its calculation and help us understand what it tells us. Observe that you must compute the ____ before you can compute the variance.

expected value of X

Where ____ indexes the possible outcomes

i

The expected value is the "theoretical" average that is computed by weighting each outcome by its _____ and then summing over all possible outcomes.

probability

We frequently summarize information for a discrete random variable by means of a _____

probability histogram

Probabilities can be represented as ____ or ____

proportions (0 to 1) or as percentages (0% to 100%).

If an uncertain event has numeric outcomes (for example, the rate of return on an asset, the selling price of a commercial or residential property, the daily/weekly demand for a product or service, the daily production of plant, a customer's satisfaction score, etc.) we call it a ____

random variable

If we have a complete description of both, then we know the _____

random variable's distribution

Note: The term _____ refers to the probability of a defect or error in a production process.

six sigma

The variance of a random variable is not the only measure of dispersion for a random variable. An easier measure to interpret is the _____, which is the square root of the variance

standard deviation

We use probabilities to talk about uncertain events. Probability is defined as ____

the chance, likelihood, or possibility that a particular event will occur.

If measured with infinitesimal precision, one could not list all of the possible outcomes. In practice, however, continuous distributions are often used to approximate discrete random variables if _____

the number of possible outcomes is very large.

The probability histogram is a visual display of ____

the outcomes (plotted along the x-axis), and the probabilities of those outcomes, which are represented by vertical bars.

For example, a manufacturer of children's products might want to know the probability that an American family has children aged 5 or younger. Using the most recent US census, you can divide the number of families with children aged 5 or younger by ____

the total number of families.

A financial analyst might want to know the probability that Apple's stock price increases after a new product introduction. Using Apple's stock price history, the analyst can divide the number of times that the stock has increased after a new product introduction by ____

the total number of new product introductions.

Over 99.5% of all observed outcomes will be _____

values that are within three standard deviations (±3σ) of the mean.