STATS EXAM 2: CHAPTER 5 Continuous Random Variables

Recognize the shape of a continuous probability distribution when it is graphed

- uniform distribution (312) -exponential distribution (313) - normal distribution (313)

Describe how an exponential probability distribution is similar to and different from other types of distributions

-Normal distribution -Geometric distribution / ehh -Poisson distribution: (336) time that elapses between two successive events follows the exponential distribution with a mean of mu units of time. Also assume that these times are independent, meaning that the time between events is not affected by the times between previous events. If these assumptions hold then the number of events per unit time follows a poisson distribution with mean lambda = 1/mu. If X has the poisson distribution with mean lambda, then P(X = K ) = lambda ^k e^-lambda/ k!. Conversely if the number of events per unit time follows a poisson distribution, then the amount of time between events follows the exponential distribution. (k! = K*(k-1*)(k-2)*(k-3)...3*2*1)

Recognize what the "area under the curve" represents in a continuous probability distribution

312 Given by a different function called the cumulative distribution function (abbreviated as CDF) The cumulative distribution function is used to evaluate probability as area.

Describe the characteristics of a cumulative distribution function (CDF)

312 Given by a different function called the cumulative distribution function (abbreviated as CDF) The cumulative distribution function is used to evaluate probability as area. -the outcomes are measured, not counted -the entire area under the curve and above the x-axis is equal to one -probability is found for intervals of x values rather than for individual x values. -P(c < x < d) is the probability that the random variable X is in the interval between the values c and d. P(c < x < d) is the area under the curve, above the x-axis, to the right of c and the left of d. - P(x = c) = 0 the probability that x takes on any single individual value is zero. the area below the curve, above the x-axis, and between x-c and x-c has no width, and therefore no area (area = 0). since the probability is equal to the area, the probability is also zero. -P(c <x< d) is the same as P(c < x < d) because probability is equal to area.

Recognize and interpret continuous probability density functions

312 Properties of Continuous probability distributions: -Graph is a curve, probability is represented by area under the curve - the curve is called the probability density function (PDF). -We use the symbol f(x) to represent the curve. PROBABILITY = AREA

Recognize the notion for the uniform distribution

324 -concerned with events that are equally likely to occur. -be careful to note if the data is inclusive or exclusive of endpoints

Recognize and use the exponential probability distribution

326) -often concerned with the amount of time until some specific event occurs. -For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. -occur the following way -there are fewer large values and more small values -commonly used in calculations of product reliability, or the length of time a product lasts

Describe the "memory less property" of the exponential probability distribution

334 Glossary: for an exponential random variable X, the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. this means that the probability that knowledge about it. In symbols we say that P(X > x + k|X > x) = P(X >k).

Calculate the probability density function when given a sample mean and sample standard deviation

Calculation the probability density function when given a sample mean and sample standard deviation PDF: probability density function f(x) represent the curve. Use the density function f(x) to draw the graph of the probability distribution f(x) area- area between it and the x-axis is equal to probability. Since maximum probability is one, the maximum area is also one. PROBABILITY = AREA Which one? Probability density function (pdf) f(x): f(x) > 0 Total area under the curve f(x) is one. Which one? This one*** Probability density function: f(x) = 1/b-a for a < X < b (this formula******)

Recognize and use the uniform probability distribution

Describe the characteristics of a uniform probability distribution

Describe the "decay parameter"

Glossary: Describes the rate at which probabilities decay to zero for increasing values of x. It is the value m in the probability density function f(x) = me^(-mx) of an exponential random variable. It is also equal to m = 1/mu, where mu is the mean of the random variable

Define a continuous probability distribution function

Use function notation f(x). Intermediate Algebra: define function so that the area between it and the x-axis is equal to a probability. since the maximum probability is one, the maximum area is also one. PROBABILITY = AREA