Let $X ~ N (4, 16)$ $X ~ N(4, 16)$ . How do you find $P (X < 6)$ $P(X < 6)$ ?

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Answer 1 · 2017-01-18T00:22:31

Use the function $Phi(x)$ and a Normal distribution table

$X~N(4,16) => P(X<4)=Phi(0.5)approx.6915$

If $X$ is a Normal Random Variable

$X~N(mu,sigma^2)$

Then its distribution function is the Gaussian Phi function

$=> P(X<=x)=F_X(x)=int_RRe^(-((x-mu)/(sigma))^2/2)dx=Phi((x-mu)/sigma)$

In this case $mu=4$ and $sigma=4$

So,

$P(X<=x)=Phi((x-4)/4)$

Note:

${X<=x}={X < x}cup{X=x}$ , and ${X < x} nn {X=x}=O/$

So,

$P(X<=x)=P(X < x)+P(X=x)$

and since a normal random variable is continuous $P(X=x)=0$

Therefore

$P(X<=x)=P(X < x)$ in this case

Because of this we can say

$P(X<6)=P(X<=6)=Phi((6-4)/4)=Phi(2/4)=Phi(0.5)$

Then we check our normal distribution tables and see that

$P(X<6)=Phi(0.5)approx.6915$

Let X ~ N(4, 16)X~N(4,16)X ~ N(4, 16). How do you find P(X < 6)P(X<6)P(X < 6)?