As disputed in section 4.1 \"Random Variables\" in chapter 4 \"Discrete random Variables\", a random variable is called *continuous* if its set of feasible values consists of a whole interval that decimal numbers. In this chapter we investigate such random variables.

You are watching: For a continuous random variable x, the probability density function f(x) represents

### Learning Objectives

To learn the concept of the probability distribution of a consistent random variable, and how the is used to compute probabilities. To learn straightforward facts around the family members of normally distributed random variables.## The Probability distribution of a constant Random Variable

For a discrete arbitrarily variable *X* the probability that *X* assumes among its feasible values on a single trial the the experiment makes great sense. This is no the instance for a continuous random variable. Because that example, intend *X* denotes the size of time a commuter just arriving at a bus stop needs to wait for the next bus. If buses operation every 30 minutes without fail, climate the set of feasible values that *X* is the term denoted <0,30>, the set of all decimal numbers in between 0 and 30. However although the number 7.211916 is a feasible value of *X*, over there is little or no meaning to the ide of the probability that the commuter will certainly wait exactly 7.211916 minutes because that the following bus. If anything the probability should be zero, due to the fact that if we could meaningfully measure the waiting time to the nearest millionth that a minute that is practically inconceivable that us would ever get *exactly* 7.211916 minutes. An ext meaningful inquiries are those that the form: What is the probability that the commuter\"s wait time is less than 10 minutes, or is between 5 and also 10 minutes? In various other words, with continuous random variables one is involved not with the event that the variable assumes a solitary particular value, yet with the event that the arbitrarily variable suspect a worth in a particular interval.

### Definition

*The* **probability circulation of a consistent random variable** *X* *is one assignment the probabilities to intervals of decimal numbers utilizing a function* f(x), *called a* thickness functionThe role f(x) such that probabilities of a constant random variable *X* are locations of areas under the graph that y=f(x)., *in the adhering to way: the probability that* *X* *assumes a value in the interval* *is same to the area that the region that is bounded over by the graph that the equation* y=f(x), *bounded below by the* *x**-axis, and bounded ~ above the left and right through the vertical lines through* *a* *and* *b*, *as shown in number 5.1 \"Probability provided as Area that a region under a Curve\"*.

Figure 5.1 Probability provided as Area of a region under a Curve

This meaning can be interpreted as a herbal outgrowth the the discussion in section 2.1.3 \"Relative Frequency Histograms\" in thing 2 \"Descriptive Statistics\". Over there we witnessed that if we have actually in check out a population (or a very big sample) and also make dimensions with greater and greater precision, then together the bars in the loved one frequency histogram become exceedingly fine your vertical political parties merge and disappear, and also what is left is simply the curve created by your tops, as shown in number 2.5 \"Sample Size and Relative Frequency Histograms\" in chapter 2 \"Descriptive Statistics\". Additionally the full area under the curve is 1, and also the ratio of the populace with measurements in between two numbers *a* and also *b* is the area under the curve and also between *a* and also *b*, as displayed in number 2.6 \"A an extremely Fine relative Frequency Histogram\" in chapter 2 \"Descriptive Statistics\". If we think that *X* as a measure to infinite precision occurring from the an option of any kind of one member the the populace at random, climate P(aXb) is simply the relationship of the population with measurements in between *a* and also *b*, the curve in the family member frequency histogram is the density role for *X*, and we come at the an interpretation just above.

Every density role f(x) must meet the adhering to two conditions:

For every numbers

*x*, f(x)≥0, so the the graph of y=f(x) never drops below the

*x*-axis. The area the the an ar under the graph of y=f(x) and above the

*x*-axis is 1.

Because the area that a line segment is 0, the meaning of the probability distribution of a constant random variable implies that because that any particular decimal number, to speak *a*, the probability that *X* suspect the precise value *a* is 0. This property indicates that whether or not the endpoints of one interval are included makes no difference worrying the probability the the interval.

### Example 1

A random variable *X* has actually the uniform distribution on the interval <0,1>: the density function is f(x)=1 if *x* is between 0 and 1 and also f(x)=0 for all various other values of *x*, as displayed in number 5.2 \"Uniform distribution on \".

Figure 5.2 Uniform circulation on <0,1>

uncover

*P*(

*X*> 0.75), the probability the

*X*suspect a value higher than 0.75. Uncover

*P*(

*X*≤ 0.2), the probability the

*X*presume a value less than or same to 0.2. Uncover

*P*(0.4

Solution:

Figure 5.3 Probabilities native the Uniform circulation on <0,1>

### Example 2

A male arrives at a bus prevent at a arbitrarily time (that is, with no regard because that the reserved service) to catch the next bus. Buses operation every 30 minutes there is no fail, thus the following bus will come any time throughout the following 30 minutes with evenly dispersed probability (a uniform distribution). Discover the probability the a bus will certainly come within the next 10 minutes.

Solution:

The graph the the density duty is a horizontal line above the interval from 0 to 30 and also is the *x*-axis anywhere else. Because the total area under the curve must be 1, the elevation of the horizontal line is 1/30. See figure 5.4 \"Probability of wait At most 10 Minutes for a Bus\". The probability sought is P(0≤X≤10). By definition, this probability is the area of the rectangular region bounded over by the horizontal heat f(x)=1∕30, bounded below by the *x*-axis, bounded top top the left by the vertical line at 0 (the *y*-axis), and bounded on the ideal by the vertical line at 10. This is the shaded an ar in number 5.4 \"Probability of wait At most 10 Minutes for a Bus\". That area is the basic of the rectangle time its height, 10·(1∕30)=1∕3. Hence P(0≤X≤10)=1∕3.

Figure 5.4 Probability of wait At many 10 Minutes for a Bus

## Normal Distributions

Most civilization have heard of the “bell curve.” it is the graph that a specific density function f(x) that explains the behavior of constant random variables as different as the heights of person beings, the quantity of a product in a container that was filled by a high-speed pack machine, or the velocities of molecule in a gas. The formula because that f(x) includes two parameters *μ* and *σ* that deserve to be assigned any specific numerical values, so lengthy as *σ* is positive. We will certainly not need to understand the formula because that f(x), but for those who space interested it is

where π≈3.14159 and also *e* ≈ 2.71828 is the basic of the natural logarithms.

Each different selection of particular numerical values for the pair *μ* and also *σ* provides a various bell curve. The worth of *μ* identify the location of the curve, as shown in number 5.5 \"Bell Curves v \". In each situation the curve is symmetric about *μ*.

Figure 5.5 Bell Curves with *σ* = 0.25 and Different worths of *μ*

The worth of *σ* determines whether the bell curve is tall and also thin or short and squat, subject constantly to the problem that the full area under the curve be same to 1. This is presented in number 5.6 \"Bell Curves through \", wherein we have actually arbitrarily chosen to facility the curves at *μ* = 6.

### Definition

*The probability distribution equivalent to the density function for the bell curve through parameters* *μ* *and* *σ* *is dubbed the* typical distributionAssignment the probabilities to a consistent random variable using a bell curve because that the thickness function. *with mean* *μ* *and standard deviation* *σ*.

### Definition

*A consistent random variable whose probabilities are explained by the normal circulation with mean* *μ* *and typical deviation* *σ* *is called a* **normally distributed random variable, or a** typical random variableA consistent random variable whose probabilities are established by a bell curve. *for short, with mean* *μ* *and typical deviation* *σ*.

Figure 5.7 \"Density role for a Normally spread Random change with average \" shows the density role that identify the normal circulation with median *μ* and also standard deviation *σ*. Us repeat an essential fact around this curve:

Figure 5.7 Density function for a Normally distributed Random variable with mean *μ* and Standard Deviation *σ*

### Example 3

Heights the 25-year-old men in a certain an ar have median 69.75 inches and also standard deviation 2.59 inches. These heights are roughly normally distributed. Thus the height *X* that a randomly selected 25-year-old man is a typical random change with mean *μ* = 69.75 and also standard deviation *σ* = 2.59. Map out a qualitatively accurate graph that the density function for *X*. Discover the probability that a randomly selected 25-year-old male is more than 69.75 customs tall.

Solution:

The circulation of heights looks favor the bell curve in figure 5.8 \"Density function for Heights of 25-Year-Old Men\". The important allude is that it is focused at that mean, 69.75, and also is symmetric about the mean.

Since the full area under the curve is 1, by the opposite the area to the right of 69.75 is fifty percent the total, or 0.5. However this area is specifically the probability *P*(*X* > 69.75), the probability that a randomly selected 25-year-old guy is an ext than 69.75 customs tall.

We will certainly learn just how to compute various other probabilities in the following two sections.

### Key Takeaways

for a continuous random change*X*the just probabilities that room computed room those of

*X*taking a worth in a stated interval. The probability the

*X*take it a worth in a details interval is the same whether or no the endpoints of the interval space included. The probability P(aXb), the

*X*take it a worth in the interval indigenous

*a*come

*b*, is the area that the region between the upright lines through

*a*and

*b*, above the

*x*-axis, and also below the graph that a duty f(x) called the density function. A normally distributed random change is one who density role is a bell curve. Every bell curve is symmetric around its mean and also lies everywhere over the

*x*-axis, which it approaches asymptotically (arbitrarily closely without touching).

A continuous random variable *X* has a uniform distribution on the term <5,12>. Lay out the graph the its density function.

A consistent random change *X* has a uniform circulation on the expression <−3,3>. Map out the graph that its density function.

A constant random change *X* has a normal circulation with mean 100 and also standard deviation 10. Map out a qualitatively exact graph that its density function.

A continuous random variable *X* has actually a normal distribution with mean 73 and also standard deviation 2.5. Lay out a qualitatively specific graph that its density function.

A consistent random change *X* has actually a normal circulation with mean 73. The probability the *X* bring away a value greater than 80 is 0.212. Use this information and also the the contrary of the density role to discover the probability the *X* bring away a value less than 66. Lay out the thickness curve through relevant regions shaded to show the computation.

A consistent random variable *X* has a normal circulation with average 169. The probability that *X* takes a value higher than 180 is 0.17. Use this information and the symmetry of the density role to discover the probability that *X* takes a value much less than 158. Lay out the density curve with relevant regions shaded to show the computation.

A continuous random variable *X* has actually a normal circulation with mean 50.5. The probability the *X* takes a value less than 54 is 0.76. Usage this information and the the contrary of the density function to uncover the probability the *X* takes a value greater than 47. Map out the thickness curve through relevant regions shaded to highlight the computation.

A constant random variable *X* has a normal distribution with average 12.25. The probability the *X* bring away a value less than 13 is 0.82. Usage this information and also the symmetry of the density duty to discover the probability that *X* takes a value higher than 11.50. Lay out the thickness curve with relevant regions shaded to highlight the computation.

The figure provided shows the thickness curves of three normally distributed random variables *XA*, *XB*, and also *XC*. Their traditional deviations (in no particular order) room 15, 7, and also 20. Usage the number to recognize the values of the means μA, μB, and also μC and also standard deviations σA, σB, and also σC of the 3 random variables.

The figure noted shows the thickness curves of three normally dispersed random variables *XA*, *XB*, and *XC*. Their traditional deviations (in no certain order) space 20, 5, and also 10. Usage the figure to identify the worths of the method μA, μB, and also μC and standard deviations σA, σB, and also σC of the three random variables.

Dogberry\"s alert clock is battery operated. The battery might fail through equal probability at any type of time that the job or night. Every day Dogberry set his alarm because that 6:30 a.m. And goes come bed at 10:00 p.m. Uncover the probability that as soon as the clock battery ultimately dies, the will do so at the most inconvenient time, between 10:00 p.m. And 6:30 a.m.

Buses running a bus line near Desdemona\"s home run every 15 minutes. There is no paying fist to the schedule she walks to the nearest prevent to take the bus come town. Find the probability that she waits much more than 10 minutes.

The amount *X* that orange juice in a randomly selected half-gallon container varies according to a normal circulation with median 64 ounces and also standard deviation 0.25 ounce.

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*X*. What ratio of all containers contain much less than a fifty percent gallon (64 ounces)? Explain. What is the average amount the orange juice in such containers? Explain.

The weight *X* that grass particle in bags significant 50 lb varies according to a normal circulation with typical 50 lb and standard deviation 1 oz (0.0625 lb).

*X*. What proportion of all bags weigh less than 50 pounds? Explain. What is the mean weight of such bags? Explain.