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.