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I'm confused on how you approach these types of questions. I never know what to start with when I have 2 random variables that are related.

Suppose that X is a random variable which takes only two values −2 and +2, and that Y = aX + b where a and b are positive constants.

If Y ∼ Bernoulli(1/2), find a and b.

Tingo Hugo
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  • Yes I know the answers but I want to know how to approach these questions next time I'm asked – Tingo Hugo Jan 02 '23 at 14:17
  • Hint: $Y$ can only be either $0$ or $1$. What can $a X +b$ be? – mowzorn Jan 02 '23 at 14:36
  • well, aX + b = 0 or aX + b = 1, but where does X's values come in? When Y is 1 X can be -2 or 2 and the same for Y = 0. I don't see it – Tingo Hugo Jan 02 '23 at 15:08
  • Let's assume that when $X=-2$ then $Y=0$. It means that $-2a+b=0$ and $2a+b=1$. Solving it gives us $a=1/4$ and $b=1/2$. Check what happens in second case. – mowzorn Jan 02 '23 at 16:18
  • @mowzorn but how do you know to equate -2a + b = 0 and 2a + b = 1 instead of 0 = -2a + b and 0 = 2a + b for example? – Tingo Hugo Jan 02 '23 at 21:50
  • I don't; in fact I did both of these cases and just explained one, but asked to check what happens in the second one. After doing both cases we see that only one solution is valid, because we have a constraint that $a$ and $b$ are positive. – mowzorn Jan 02 '23 at 21:53
  • @mowzorn oh okay thanks! – Tingo Hugo Jan 02 '23 at 21:56

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