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

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$\mu = 2$

$\sigma^2 = 25$

$\sigma = 5$

$Z_1 =$$\large \frac{1 - 2}{5} = -0.2 Z_2 =$$\large \frac{4 - 2}{5} = 0.4$

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

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

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

$P (1 < X < 4) = 0.2347$
by Boss (36.7k points)
edited
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What is phi(0.2) ? and how you know the Z of (0.2) without table ( without external source except given in question ) ?

+1

there was some mistakes i corrected them

$\phi(Z)$ means the the Cumulative distribution function.

value of $\phi$ must be given, actually i'm surprised that how in 2010 it is not given.

So what is the intuition behind this question?

check the shaded area we calculate the area between 1 and 4.

but in exam only $\phi(0.4)$ will be given and what does that mean?

It means area to the left of 4.

Next what does $\phi(-0.2)$ gives us? It gives us the area left of 1 but we don't need that, we need area right of 1 and left of 4. (Area in first image).

So we know $\phi(-0.2) = 0.4207$ and $\phi(0.4) = 0.6554$

So we can subtract the area left of 1 from area left of 4.

0.6554 - 0.4207 gives us 0.2347

+2
I used the identity $\phi(-Z) = 1 - \phi(Z)$

so that incase the value of $\phi(Z)$ is given then also one can solve this question using this identity
+2

What's the use of calculating $Z$ in normal distribution?

While calculating Z we convert any normal distribution to Standard Normal Distribution.

Standard Normal Distribution : a mean of 0 and a standard deviation of 1

So when value of Z is $0.4$ then $\phi(0.4)$ actually means area to the left of 0.4 where mean is 0 and standard deviation is 1.

+1

Hey, its really a beautiful explanation , it cleared my concepts and added some extra too. Thanx ☺️

But raised some doubts :

1. Is it the given area for particular Z value always always a area left of it ?

2. If so, then no worries , we can manipulate the area we wanted by using property that you given ( as its equal both the sides of Z right ? ),  But if not , if its not always left part then for given value  ,say  given is Z(1.4)=0.432    then what to consider, left or right  i mean how to decide ? ( is it depends on our given interval ?)

3. In given context , two values are mandatory to be mentioned in exam i.e z(0.2) and z(0.4) right ? or we can somehow do it with only one value given ? ( i guess we can not without mentioning it ? )

+1
1. Cumulative distribution function: a function whose value is the probability that a corresponding continuous random variable has a value less than or equal to the argument of the function. CDF is represented by $\phi$. So yes if $\phi$. is given then that means area to the left of Z.
2. I think 1st point cleared your 2nd point too.
3. For this question 2 values are mandatory without it one cannot answer this question in given time. Also i never calculated CDF from formula given here.
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Pretty cool. Thanks :)

yes its tedious to do so using pdf of normal distribution.

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@Mk Utkarsh

Above you used :

Z= (Mu - X)/ Sig

Z = ( X- Mu)/sig

can they be used interchangeably ?

P.S : Okay , sorry.. you corrected it.

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yes that was a mistake
+1

Yes. Thank you :)

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@Mk Utkarsh

Any reference to the statements that CDF always less than given argument. And Z is always taken as left.. So that i can read from that.

+1
Sheldon Ross
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@Mk Utkarsh

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.

+1

look 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

–1 vote
Is it 0.024 ?
by Active (2.5k points)
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think logically, can probability be so low if the mean is 2?
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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 ?