standard normal distribution

[gaia00]

hi! i'm new here so I hope this is posted correctly..
they are two similar simple true/false question on standard normal distribution:

Let Z ~ N=(0,1) then P(|Z| ≥ 2) = 2*P(Z ≥ 2). is this equality true or false?
Let Z ~ N=(0,1) then P(-1≤Z≤0) = 0.5*P(-1≤Z≤1). again, true or false?

I think they are both true, since normal distribution is symmetric but I just want to make sure I'm not missing out on something else.. in that case I kindly ask for a brief explanation if possible.

thanks in advance ))))

 

[Dr.Peterson]

hi! i'm new here so I hope this is posted correctly..
they are two similar simple true/false question on standard normal distribution:

Let Z ~ N=(0,1) then P(|Z| ≥ 2) = 2*P(Z ≥ 2). is this equality true or false?
Let Z ~ N=(0,1) then P(-1≤Z≤0) = 0.5*P(-1≤Z≤1). again, true or false?

I think they are both true, since normal distribution is symmetric but I just want to make sure I'm not missing out on something else.. in that case I kindly ask for a brief explanation if possible.

thanks in advance ))))

Yes, both are correct because of symmetry.

To prove it more explicitly, consider the first:

P(|Z| ≥ 2) = P(Z ≥ 2 or Z ≤ -2) = P(Z ≥ 2) + P(Z ≤ -2) =P(Z ≥ 2) + P(Z ≥ 2) = 2P(Z ≥ 2)​