A random variable X is normally distributed with mu=60 and sigma = 3. What is the value of 2 numbers a,b so that P(X=a)=P(X=b). The solution is a=60 and b=65. However, I do not know how to come up with that answer. As far as I understand P(X=a) and P(X=b) have to be both 0 since you always have to give a range e.g. P(a<X). Moreover if I insert the values 60 and 65 in the formula Z=(X−mu)/sigma than I would end up with 0,1.667 and z-scores 0.5, 0.952 respectively.

Mattie Monroe 2022-10-21 Answered
A random variable X is normally distributed with μ = 60 and σ = 3. What is the value of 2 numbers a,b so that P ( X = a ) = P ( X = b ).
The solution is a = 60 and b = 65.
However, I do not know how to come up with that answer. As far as I understand P ( X = a ) and P ( X = b ) have to be both 0 since you always have to give a range e.g. P ( a < X ). Moreover if I insert the values 60 and 65 in the formula Z = ( X μ ) / σ than I would end up with 0,1.667 and z-scores 0.5, 0.952 respectively.
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Answers (2)

Audrey Russell
Answered 2022-10-22 Author has 16 answers
Your reasoning is valid -- asking for P ( X = a ) = P ( X = b ) makes essentially no sense because such a probability is always 0 for a continuous distribution.
Indeed, this seems to be the only way to make a = 60 , b = 65 a valid answer.
One might try to be charitable and "correct" the question into asking for two points where the probability density is the same -- but that wouldn't lead to a = 60 , b = 65 being a solution; instead we would have a = 60 + t , b = 60 t for some t (since the distribution is symmetric around μ = 60).
My tentative conclusion would be that (a) it's a trick question, (b) your understanding is correct, and (c) the solution you quote is just meant to be one possible answer.
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Evelyn Freeman
Answered 2022-10-23 Author has 5 answers

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