Problem with conditional probabilities associated with random variables as well as hypotheses

Question

I need urgent help with a problem I have been considering since two weeks. Namely, this problem is with conditional probabilities.
Let $X=\mu+W$ , $W\sim\mathcal{N}(0,1)$, $\mu=\gamma Z$ where $W$, $Z$ and $\gamma$ are independent random variables, $Z\sim\mathcal{N}(0,3)$, $P(\gamma=0)=1/5,P(\gamma=1)=4/5.$ I have a problem with evaluating two probabilities: $P(\mu=0|X)$ and $P(H_0 \ \textrm{accepted}|H_0 \ \textrm{false})$ when we know that $H_0: \mu=0 $ is rejected when $P(\mu=0|X)<1/2$.
My attempts:
$P(\mu=0|X)=\dfrac{P(\mu=0, X)}{?}=\dfrac{P(\mu=0, \mu+W)}{?}=\dfrac{P(W)}{?}=?$ $P(H_0 \ \textrm{accepted}|H_0 \ \textrm{false})=\dfrac{P(H_0 \ \textrm{accepted and}\ H_0 \ \textrm{false})}{P(H_0 \ \textrm{false})}=\dfrac{?}{P(\gamma Z=1)}=?$
Unfortunately I don't know what should I do in the next steps. I would really appreciate any help and advices. Thanks in advance!

Answer 1

The difference between the events $\mu = 0$ and $\gamma = 0$ has probability $0$, so we may equivalently ask for $P(\gamma = 0 \mid X)$. This may be obtained from considering $P(\gamma = 0 \mid x \le X \le x + \delta)$ as $\delta \to 0+$. Let $f(x) = e^{-x^2/2}/\sqrt{2\pi}$ be the standard normal PDF. If $\gamma = 0$, $X = W \sim N(0,1)$, while if $\gamma = 1$, $X = W+Z \sim N(0,\sqrt{1+3^2}=\sqrt{10})$. Thus $$P(\gamma = 0, x \le X \le x + \delta) = P(\gamma=0, x \le W \le x + \delta) = \frac{1}{5} f(x) \delta + o(\delta)= \frac{\delta}{5 \sqrt{2\pi}} e^{-x^2/2} + o(\delta)$$ while since $(W+Z)/\sqrt{10}$ is standard normal, $$\eqalign{P(\gamma = 1, x \le X \le x + \delta) &= P(\gamma=1, x \le W+Z \le x+\delta)\cr &= P\left(\gamma = 1, \frac{x}{\sqrt{2}} \le \frac{W+Z}{\sqrt{10}} \le \frac{x+\delta}{\sqrt{10}}\right)\cr& \frac{4}{5} f\left(\frac{x}{\sqrt{10}}\right) \frac{\delta}{\sqrt{10}} + o(\delta) = \frac{\delta}{10 \sqrt{2\pi} }e^{-x^2/200} + o(\delta)}$$ Thus $$ \eqalign{ P(\gamma = 0 \mid x \le X \le x+\delta) &= \frac{P(\gamma = 0, x \le X \le x + \delta)}{P(\gamma = 0, x \le X \le x + \delta) + P(\gamma = 1, x \le X \le x + \delta) }\cr &\approx \frac{1}{1 + 2 \sqrt{2} e^{x^2/4}}}$$ Thus we should have $$P(\mu=0 \mid X) = \frac{5}{5 + 2 \sqrt{2}\; e^{99 X^2/200}}$$

We should accept $H_0$ if this $> 1/2$, i.e. $|X| < 10 \sqrt{22 \ln(5/2)}/33$. Can you do the rest?