4. Random Variables A random variable is a way of recording a quantitative variable of a random...

4. Random Variables

A random variable is a way of recording a quantitative variable of a random experiment.

4. Random Variables


In particular Chapter 4 talks about discrete random variables.

4. Random Variables



If a random variable has a particular distribution (such as a binomial distribution) then our work becomes easier. We use formulas and tables.

5. Continuous Random Variables



If a random variable has a particular distribution (such as a binomial distribution) then our work becomes easier. We use formulas and tables.

5. Continuous Random Variables

A random variable where X can take on a range of values, not just particular ones.

Examples:

Heights

Distance a golfer hits the ball with their driver

Time to run 100 meters

Electricity usage of a home.

Continuous probability distribution functions

For a discrete random variable, probabilities are given as a table of values, and the distribution can be graphed as a bar graph.

For a continuous random variable, probabilities are specified by a continuous function. The graph of the probability distribution function is a curve.

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Figure 5.1 A probability f(x) for a continuous random variable x


Definition


Figure 5.2 Density Function for Friction Coefficient, Example 5.1


Find probability friction is less than 10.


Find probability friction is less than 10.Solution:Probabiity = area of shaded

triangle = (1/2)(5)(0.2)=0.5

Uniform Distribution

A Uniform Distribution has equally likely values over the range of possible outcomes.

Uniform Distribution

A Uniform Distribution has equally likely values over the range of possible outcomes.

A graph of the uniform probability distribution is a rectangle with area equal to 1.

Example The figure below depicts the probability distribution for

temperatures in a manufacturing process. The temperatures are controlled so that they range between 0 and 5 degrees Celsius, and every possible temperature is equally likely.

x

P(x)

Temperature (degrees Celsius)

0 1 2 3 4 5

0.2

0

Example Note that the total area under the

“curve” is 1.

x

P(x)


0 1 2 3 4 5

0.2

0

Example

What is the Probability that the temperature is exactly 4 degrees?

x

P(x)


0 1 2 3 4 5

0.2

0

Example

What is the Probability that the temperature is exactly 4 degrees?

Answer: 0

x

P(x)


0 1 2 3 4 5

0.2

0

Since we have a continuous random variable there are an infinite number of possible outcomes between 0 and 5, the probability of one number out of an infinite set of numbers is 0.

Explanation

Example

What is the probability the temperature is between 10C and 40C?

x

P(x)


0 1 2 3 4 5

0.2

0

What is the probability the temperature is between 10C and 40C?

We know that the total area of the rectangle is 1, and we can see that the part of the rectangle between 1 and 4 is 3/5 of the total, so P(1 x 4) = 3/5*(1) = 0.6.

x

P(x)


0 1 2 3 4 5

0.2

0

Review: Probabilities and Area For a density curve depicting the

probability distribution of a continuous random variable, – the total area under the curve is 1, – there is a direct correspondence between area

and probability.– Only the probability of an event occurring in

some interval can be evaluated. – The probability that a continuous random

variable takes on any particular value is zero.

General Uniform Distribution

A Uniform Distribution has equally likely values over the range of possible outcomes, say c to d.

12Deviation Standard

2Mean

1 :functiondensity theofHeight

cd

dccd

f(x)

Normal Distributions

This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bell-shaped curves.


This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bell-shaped curves.

2

/ 22 2)(

xe

y

Reminder: Mu is the mean, sigma is the standard deviation.

Examples

The following are examples of normally distributed everyday data.

– Grades on a test.– How many chips are in a small bag of potatoe

chips.– The measurements of distance between two

points.– The heights of students in this class.


Shape of this curve is determined by µ and σ – µ it’s centered, σ is how far it’s spread out.

Standard Normal Distribution

The Standard Normal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

In this way the formula giving the heights of the normal curve is simplified greatly.

Z-score

Standard Normal Probabilities

P(0 z 1) represents the probability that z takes on values between 0 and 1, which is represented by the area under the curve between 0 and 1.

P(0 z 1) = 0.3413

P(0 z 1) = 0.341

Revelation!

Since the mean is 0 and the standard deviation is 1, this tells us that the probability that z is within one standard deviation of the mean (either below or above) is (2)(0.341)= 0.682.

P(0 z 1) = 0.341

Revelation!

Since the mean is 0 and the standard deviation is 1, this tells us that the probabiity that z is within one standard deviation of the mean (either below or above) is (2)(0.341)= 0.682.

Agrees with Empirical Rule: 68% of data lies within one standard deviation of the mean

Finding Probabilities when given z-scores.

For a given z-score, the probability can be found in a table in the back of the text (Table IV), also see inside front cover.

Note: The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks.


Table 5.1


Find probability z is between -1.33 and +1.33.


Want probability z is between -1.33 and +1.33.Solution: Locate 1.33 in the row labeled 1.3 and the column labeled .03. By symmetry, ans = 2(0.4082) = .8164


Find probability z exceeds 1.96 in absolute value.


Areas under the standard normal curve for z exceeding 1.96 in absolute value

Revelation!

It follows that the area of the un-shaded region is 0.95. Agrees with Empirical Rule which states that,

for data sets having a mound shaped distribution, 95% of the

values lie within approximately 2 standard deviations of the mean

Keys to success

Learn the standard normal table and how to use it.

We will be using these tables through out the course.

4. Random Variables A random variable is a way of recording a quantitative variable of a random...

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