Probability- Gravner- 76

Question

Assume that $X$ is Normal with mean $\mu$ $=$ $2$ and variance $\sigma^2$ $=$ $25$. Compute the probability that $X$ is between $1$ and $4$.

Answer 1

$\mu = 2$

$\sigma^2 = 25$

$\sigma = 5$

$Z_1 =$$\large \frac{1 - 2}{5} = -0.2$

$Z_2 =$$\large \frac{4 - 2}{5} = 0.4$

$P (1 < X < 4) = \phi(0.4) - \phi (-0.2)$

$P (1 < X < 4) =  \phi(0.4) - (1 - \phi (0.2))  $

$P (1 < X < 4) = 0.6554  - (1 - 0.5793)$

$P (1 < X < 4) = 0.2347$

Comments

Sheldon Ross

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@Mk Utkarsh

I have downloaded two books :

1.Introduction to Probability Models Ninth Edition

2. AFIRST COURSE IN PROBABILITY Eighth Edition

Actually i amanot getting where exact point of CDF and related "Z" info is , if you know where it is please mention.

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look Z is standard normal random variable. Calculation of Z changes the question to standard normal distribution and sets the mean to 0 and standard deviation to 1. 

For more details on CDF read definition here

Answer 2

Is it 0.024 ?

Comments

think logically, can probability be so low if the mean is 2?

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@Mk Utkarsh 

Hey, 

If any how if restrict the Z value , 0<=Z<=0.01 

then wouldn't be the probability be less even though mean is 2 ?

Explain the statement why if mean is given something, we can so sure that probability must be greater than so and so. Please explain that logic.

Or your statement was for this question only ?