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 &lt; 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.

Question

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  &amp;lt;  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.

Audrey Russell · Accepted Answer

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 &quot;correct&quot; the question into asking for two points where the probability density is the same -- but that wouldn&#039;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&#039;s a trick question, (b) your understanding is correct, and (c) the solution you quote is just meant to be one possible answer.

Evelyn Freeman · Accepted Answer

the question is pretty meaningless

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