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Let x follow a Normal distribution with mean 2 and variance 4. This question is not a duplicate. One question is stating between two variables

Basic stats question having trouble figuring this out.

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    do you know the cumulative distribution function of normal distribution? – 2017-02-26
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    @Guangliang No,I am unfamiliar on how to get to that answer from using it – 2017-02-26
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    Please read [this post](http://meta.math.stackexchange.com/a/9960) and the others there for information on writing a good question for this site. In particular, people will be more willing to help if you [edit] your question to include some motivation, and an explanation of your own attempts. – 2017-02-26
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    Possible duplicate of [What is the probability that $x$ is between $-1.92$ and $5.92$?](http://math.stackexchange.com/questions/2162864/what-is-the-probability-that-x-is-between-1-92-and-5-92) – 2017-02-26
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    @pjs36 how if one is stating between two variables and the other just numbers before? – 2017-02-26

1 Answers 1

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Hint:

$$Pr(X < 5.92) = Pr\left(\frac{X-2}{2} < \frac{5.92-2}{2}\right)$$

and

$$\frac{X-2}{2} \sim N(0,1)$$

Edit:

Such values are usually found by using statistical tables (e.g. http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf [Credit:JMoravitz] )

$$\frac{5.92-2}{2}=1.96$$

Reading from the table, $F_Z(1.96) = 0.975 $ which is the value that you want to find.

Remark: $1.96$ is a "special" value as it is commonly used in constructing $95\%$ confidence interval.

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    What would be the X in this case then? – 2017-02-26
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    X is just a random variable – 2017-02-26
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    @user29418 Im unsure where to go with this – 2017-02-26
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    Lets call $\frac{X-2}{2}$ something else, $Z$. We have that $Z$ is the standard normal distribution with mean zero and standard deviation $1$. The value of $Pr(Z<\frac{5.92-2}{2})$ can be found with calculus, or you can simply look up the value (*or at least a nearby value*) on a table such as [this one](http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf). Any textbook with any reference to a normal curve will include such a table somewhere, often in the appendix. – 2017-02-26