Download Normal Distribution: Properties and Probabilities - Prof. Mary Flagg and more Study notes Mathematics in PDF only on Docsity! Math 1313 Section 8.5 Normal Distribution In addition to these notes, you will need to bring a copy of pages 1177 – 1178, the standard normal distribution table to class. This table is located in an appendix in the online text. In this section, we will study normally distributed random variables. These are continuous random variables. We can see the difference between finite discrete random variables and continuous random variables by looking at graphs of them side by side. 0 0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 #1 With a finite discrete random variable, we can create a table of values and list each number to which the random variable assigns a value. With a continuous random variable, we can’t do that. There are an infinite number of values. We will look at a probability density function instead. Here are properties of probability density functions: 1. for all values of x ( ) 0>f x 2. The area between the curve and the x axis is 1 Some probability density functions have additional properties. These are called normal distributions. 3. there is a peak at the mean 4. the curve is symmetric about the mean 5. 68.7% of the data is within one standard deviation of the mean; 95.45% of the area is within two standard deviations of the mean; 99.73% of the area is within three standard deviations of the mean Math 1313 Class Notes – Section 8.5, Page 1 of 7 6. the curve approaches the x axis as x extends indefinitely in either direction Some normal distributions have additional properties. These are called standard normal distributions. 7. mean is 0 8. standard deviation is 1 Notation: , so we will use ( ) ( ) ( ) (≤ ≤ = < ≤ = ≤ < = < <P a X b P a X b P a X b P a X b) )( < <P a X b as our standard notation. This is true because the area under a point is 0. Notation: We will denote the random variable which gives us the standard normal distribution by Z. Math 1313 Class Notes – Section 8.5, Page 2 of 7 Example 6: Find ( .68 1.41)− < <P Z Sometimes, we want the find z, the number inside the parentheses given a probability. IV. Find z if P(Z < z) = a particular number Look up the number in the chart. The row heading plus the column heading is z. Example 7: Find z if ( ) .8944< =P Z z Example 8: Find z if ( ) .0401< =P Z z V. Find z if P(Z > z) = a particular number This says that the area to the right of the number z is the number given. So, the area to the LEFT of –z is this probability. Look up the probability on the chart, find the row and column. The answer is the NEGATIVE of this number. Example 9: Find z if ( ) .9463> =P Z z Example 10: Find z if ( ) .0132> =P Z z There is also another version of this. Example 10.5: Find z if P(Z<-z) = .3228 Math 1313 Class Notes – Section 8.5, Page 5 of 7 VI. Find z if the probability between z and –z is given. We are given a probability, which is the area under the curve from –z to z for some number z. How to find z: Take the number given, add 1 then divide that by 2. Look this number up on the chart, the column plus row is z. Example 11: Find z if ( ) .9812− < < =P z Z z Example 12: Find z if ( ) .5408− < < =P z Z z Example 13: Find z if ( ) .1820− < < =P z Z z Next, we need to look at what to do with a distribution that is normal, but not standard normal; that is, a normally distributed random variable with mean other than 0 and standard deviation other than 1. Math 1313 Class Notes – Section 8.5, Page 6 of 7 We can convert any problem involving probability of a normally distributed random variable to one with a standard normal random variable, thus allowing us to use the table. Here’s how: Suppose X is a normal random variable with ( ) =E X μ and standard deviation = σ . Then X can be converted to the standard normal random variable using the formula − = XZ μ σ . Then we can evaluate the new problem using the techniques presented earlier in the lesson. Example 14: Suppose X is a normally distributed random variable with 50=μ and 30=σ . Find ( 95).<P X Example 15: Suppose X is a normally distributed random variable with 85=μ and 16=σ . Find ( 54>P X ). Example 16: Suppose X is a normally distributed random variable with 100=μ and 20=σ . Find (85 110).< <P X Math 1313 Class Notes – Section 8.5, Page 7 of 7
