### 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.

According to our guidelines we are solving only first three sub-parts of a question. Given that F(X…

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

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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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