### Understanding Probability Calculations Using the Z Transform for a Normal DistributionIn 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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Answered: 2) Recall F(X = t) = Pr(X < t)…

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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Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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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.

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