2) Recall F(X = t) = Pr(X < t) calculate the following using the Z transform and F(Z=t) for the random variable N(X, µ, o) a) Pr(a < X < b) b) Pr(X < b) c) Pr(a < X)
2) Recall F(X = t) = Pr(X < t) calculate the following using the Z transform and F(Z=t) for the random variable N(X, µ, o) a) Pr(a < X < b) b) Pr(X < b) c) Pr(a < X)
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
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![### Understanding Probability Calculations Using the Z Transform for a Normal Distribution
In this section, we will review the process of calculating various probabilities for a random variable \( X \) that follows a normal distribution, denoted by \( N(X, \mu, \sigma) \). The Z transform will be used to perform these calculations.
Recall the cumulative distribution function (CDF) of a random variable \( X \) given by:
\[ F(X = t) = Pr(X \leq t) \]
Using the Z transform, we will calculate the following probabilities:
#### a) \( Pr(a \leq X \leq b) \)
To find the probability that \( X \) lies between \( a \) and \( b \):
\[ Pr(a \leq X \leq b) = F(b) - F(a) \]
Where \( F(t) \) is the CDF of \( X \).
#### b) \( Pr(X \leq b) \)
To find the probability that \( X \) is less than or equal to \( b \):
\[ Pr(X \leq b) = F(b) \]
#### c) \( Pr(a \leq X) \)
To find the probability that \( X \) is greater than or equal to \( a \):
\[ Pr(a \leq X) = 1 - F(a) \]
#### d) \( Pr(|X| \leq a) \)
To find the probability that the absolute value of \( X \) is less than or equal to \( a \):
\[ Pr(|X| \leq a) = Pr(-a \leq X \leq a) = F(a) - F(-a) \]
#### e) \( Pr(|X| \geq a) \)
To find the probability that the absolute value of \( X \) is greater than or equal to \( a \):
\[ Pr(|X| \geq a) = 1 - Pr(|X| \leq a) = 1 - (F(a) - F(-a)) \]
It is important to transform the variables accordingly when using the Z transform, to standardize the normal distribution and make use of standard Z tables to find cumulative probabilities.
By understanding these calculations and transformations, we can analyze problems involving normally distributed variables more effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ae4e595-d9b4-4096-a3fc-f4bee7947b38%2F32ecd75e-46ec-469d-bcca-47ea65934225%2F3x21vvi_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Probability Calculations Using the Z Transform for a Normal Distribution
In this section, we will review the process of calculating various probabilities for a random variable \( X \) that follows a normal distribution, denoted by \( N(X, \mu, \sigma) \). The Z transform will be used to perform these calculations.
Recall the cumulative distribution function (CDF) of a random variable \( X \) given by:
\[ F(X = t) = Pr(X \leq t) \]
Using the Z transform, we will calculate the following probabilities:
#### a) \( Pr(a \leq X \leq b) \)
To find the probability that \( X \) lies between \( a \) and \( b \):
\[ Pr(a \leq X \leq b) = F(b) - F(a) \]
Where \( F(t) \) is the CDF of \( X \).
#### b) \( Pr(X \leq b) \)
To find the probability that \( X \) is less than or equal to \( b \):
\[ Pr(X \leq b) = F(b) \]
#### c) \( Pr(a \leq X) \)
To find the probability that \( X \) is greater than or equal to \( a \):
\[ Pr(a \leq X) = 1 - F(a) \]
#### d) \( Pr(|X| \leq a) \)
To find the probability that the absolute value of \( X \) is less than or equal to \( a \):
\[ Pr(|X| \leq a) = Pr(-a \leq X \leq a) = F(a) - F(-a) \]
#### e) \( Pr(|X| \geq a) \)
To find the probability that the absolute value of \( X \) is greater than or equal to \( a \):
\[ Pr(|X| \geq a) = 1 - Pr(|X| \leq a) = 1 - (F(a) - F(-a)) \]
It is important to transform the variables accordingly when using the Z transform, to standardize the normal distribution and make use of standard Z tables to find cumulative probabilities.
By understanding these calculations and transformations, we can analyze problems involving normally distributed variables more effectively.
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