DEFINITION: A normal random variable with mean μ = 0 and variance σ² called a STANDARD NORMAL random variable. 1 f(z; 0, 1): e √2πT = 1 is Notation: Z ~ Normal(0,1). The CDF of Z is P(Z ≤ 2) and will be denoted by (z). Remark: The standard normal distribution does not often arise naturally, but it rather serves as a standard against which other normal distributions are measured. Example: Computing probabilities for Normal random variables by integrating the PDF is not straightforward. Instead, tables are used which contain values of the standard normal CDF (z) for many different values of z. Let Z be a standard normal random variable. Use the table in the back of the textbook (table A3 in Devore) to find (a) P(Z ≤ 1.17) (b) P(Z-2.73) (c) P(-0.5≤ Z ≤ 0.64) Remark: Of course, you can also use software (Excel, or your calculator) to find val- ues of the standard normal CDF. In Excel, the command for (z) is Normsdist (z). Read the manual for your calculator, to find the command for (z). For example, on a TI-84, you would hit DISTR and normalcdf and enter the lower, and upper bounds and μ and σ (in that order). Poll Question 8.5 Normal Percentiles The table in the back of the book can be read backwards to find standard normal percentiles. Example: Use the table in the back of the book to find the 95th percentile of the standard normal distribution.
DEFINITION: A normal random variable with mean μ = 0 and variance σ² called a STANDARD NORMAL random variable. 1 f(z; 0, 1): e √2πT = 1 is Notation: Z ~ Normal(0,1). The CDF of Z is P(Z ≤ 2) and will be denoted by (z). Remark: The standard normal distribution does not often arise naturally, but it rather serves as a standard against which other normal distributions are measured. Example: Computing probabilities for Normal random variables by integrating the PDF is not straightforward. Instead, tables are used which contain values of the standard normal CDF (z) for many different values of z. Let Z be a standard normal random variable. Use the table in the back of the textbook (table A3 in Devore) to find (a) P(Z ≤ 1.17) (b) P(Z-2.73) (c) P(-0.5≤ Z ≤ 0.64) Remark: Of course, you can also use software (Excel, or your calculator) to find val- ues of the standard normal CDF. In Excel, the command for (z) is Normsdist (z). Read the manual for your calculator, to find the command for (z). For example, on a TI-84, you would hit DISTR and normalcdf and enter the lower, and upper bounds and μ and σ (in that order). Poll Question 8.5 Normal Percentiles The table in the back of the book can be read backwards to find standard normal percentiles. Example: Use the table in the back of the book to find the 95th percentile of the standard normal distribution.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 14E
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Transcribed Image Text:DEFINITION: A normal random variable with mean μ = 0 and variance σ²
called a STANDARD NORMAL random variable.
1
f(z; 0, 1):
e
√2πT
=
1 is
Notation: Z ~ Normal(0,1). The CDF of Z is P(Z ≤ 2) and will be denoted by
(z).
Remark: The standard normal distribution does not often arise naturally, but it
rather serves as a standard against which other normal distributions are measured.
Example: Computing probabilities for Normal random variables by integrating
the PDF is not straightforward. Instead, tables are used which contain values of
the standard normal CDF (z) for many different values of z.
Let Z be a standard normal random variable. Use the table in the back of the
textbook (table A3 in Devore) to find
(a) P(Z ≤ 1.17)
(b) P(Z-2.73)
(c) P(-0.5≤ Z ≤ 0.64)
Remark: Of course, you can also use software (Excel, or your calculator) to find val-
ues of the standard normal CDF. In Excel, the command for (z) is Normsdist (z).
Read the manual for your calculator, to find the command for (z). For example,
on a TI-84, you would hit DISTR and normalcdf and enter the lower, and upper
bounds and μ and σ (in that order).
Poll Question 8.5
Normal Percentiles
The table in the back of the book can be read backwards to find standard normal
percentiles.
Example: Use the table in the back of the book to find the 95th percentile of the
standard normal distribution.
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