Concepts and Terminology Chapter 7

If X is a continuous random variable, do P(X > 2) and P(X ≥ 2) have the same value? Explain.

Yes, all continuous probability distributions assign probability 0 to every individual outcome. Only intervals of values have positive probability.

In a probability histogram what is the sum of the heights of each bar?

Because the heights are probabilities, they add to 1.

Explain the Law of Large Numbers.

The Law of Large Numbers says that the average of the values of X observed in many trials must approach μ.

What is meant by the expected value of X ?

The expected value of a random variable X is an average of all possible values of X taking into account the fact that all values do not need to be equally likely. This expected value need not be a possible value for X.

In a probability histogram what does the height of each bar represent?

The height of each bar shows the probability of the outcome at its base.

Explain the difference between the notations x̄ and μx.

The mean x̄ is the ordinary average of a set of observations. μx is the mean, or expected value of a random variable X.

If X is a discrete random variable, what information does the probability distribution of X give?

The probability distribution for X lists the values and their probabilities. The probabilities Pi must satisfy two requirements: 1. Every probability Pi is a number 0 and 1. 2. p1 + p2 + ... + pk = 1 Find the probability of any event by adding the probabilities Pi of the particular values Xi that make up the event.

If X is a continuous random variable, how is the probability distribution of X described?

The probability distribution of X is described by a density curve.

What is the area under a probability density curve equal to?

The probability of any event is the area under the density curve and above the values of X that make up the event.

What is a continuous random variable?

A continuous random variable X takes all values in an interval of numbers.

What is a discrete random variable?

A discrete random variable X has a countable number of possible values.

What is the difference between a discrete random variable and a continuous random variable?

A discrete random variable has a countable number of possible values. A continuous random variable has an infinite number of possible values, all the values in an interval.

What is a random variable?

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

What does the density curve of a uniform distribution look like?

A uniform distribution has the same height (1/b-a), over the entire interval (a, b.)

If a normal distribution is always a probability distribution, is a probability distribution always a normal distribution?

No, not every probability distribution is a normal distribution.

If X is a discrete random variable, do P(X > 2) and P(X ≥ 2) have the same value? Explain.

No. P(X = 2) may be positive, so P(X ≥ 2) would equal P(X > 2) + P( X = 2).

How is a normal distribution related to probability distribution?

Normal distributions are one type of continuous probability distribution.

Explain how to calculate the variance of a discrete random variable X using the formula σ^2x = Σ(Xi-μx)^2 (Pi)

This formula is saying that you will take an event and subtract it from the mean to get the distance of that variable from the mean. Then we square that distance to get rid of the negative, and multiply that answer by the probability of that event occurring. You do this for each event, and then sum those answers together to get the variance.

How do you calculate the mean of a discrete random variable X?

μ = Σ (Xi)(Pi)