1

If $N(1,4)$, find $P(X<0)$ and $P(|x|>4)$. I already have the answers which are $0.3085$ and $0.07302$ respectively. please note this is not a homework. Can someone explain in detail how to reach the answer? I tried multiple times but every time I am not getting it.

Tim
  • 108,699
  • 20
  • 212
  • 390
user65652
  • 15
  • 1
  • 1
  • 5
  • 1
    This thread answers your question: https://stats.stackexchange.com/questions/19195/explaining-two-tailed-tests if for some reason it doesn't, please edit your question to clarify what is not clear. – Tim Feb 22 '16 at 15:26
  • that does not say anything about absolute value ! – user65652 Feb 22 '16 at 15:28
  • Yes it does, absolute value is the same as using two-tailed test. – Tim Feb 22 '16 at 15:31
  • my problem is how to expand the absolute value, usually we have P(14)=P(-4>X>4) , it seems very unusual to me. – user65652 Feb 22 '16 at 15:36
  • Given that this is not homework, how did this problem arise? – Glen_b Feb 23 '16 at 10:21

2 Answers2

5

In general case

$$ \Pr(|X| > x) = \Pr(X < -x \cup X > x) \\ = \Pr(X < -x) + \Pr(X > x) $$

and since normal distribution is symmetric,

$$ \Pr(X < -x) = \Pr(X > x) $$

what leads to

$$ \Pr(X < -x \cup X > x) = 2 \times \Pr(X > x) $$

You are interested in $\Pr(|X| > x)$ if you want to learn something about tails of distribution, e.g. when you want to learn about probabilities of observing values that are anomalous, outlying, or extreme. If you want to imagine this graphically, we are looking at the two tails of normal distribution (grayed).

enter image description here

You can also easily compute the opposite probability

$$ \Pr(|X| < x) = \Pr(-x < X < x) \\ = \Pr(X < x) - \Pr(X \le -x) \\ = \Pr(X > -x) - \Pr(X \ge x) \\ = 1 - \Pr(X < -x \cup X > x) $$

that is, the "common area" (colored in orange) of the two cases.

enter image description here

Check also a thread about single- vs two-tailed tests that discusses practical application of such interval probabilities.

Tim
  • 108,699
  • 20
  • 212
  • 390
1

The 'two tailed' probabilities referred to here Explaining two-tailed tests correspond to the absolute value in your question. P(abs(x)>4) is the same as P(X < - 4) + P (X > 4). Just translate X to a standard Normal variable (subtract the mean and divide by the standard deviation) and look up the corresponding probability.

Or, in R:

For the absolute value one:

pnorm(-4,mean= 1,sd = 2) + (1-pnorm(4,mean = 1,sd = 2))

For the other one:

pnorm(0,mean = 1,sd = 2)

Jordan Collins
  • 520
  • 2
  • 7