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The life span of a machine follow the law $\mathcal N(174,29^2)$.

1) On 200 machines, how many will have a life span of more that 160 days ?

2) After how many day the probability that a machine still work is less that $0.1$ ?

Attempts

I have problem to solve this two question.

1) I was thinking about set $X_i$ is 1 if the $i-$th machine work after 160 days and 0 otherwise, and try to use that central limit theorem with $S_n=X_1+...+X_n$, but I don't really see what to compute. Is it to find $S_n$ s.t. $$\mathbb P\{\frac{S_n-n\mu}{n\sigma }\geq 160\}=1\ \ ?$$

It looks weird, espacially that I don't know $\mu$ neither $\sigma $.

2) Same problem, I don't really see how to interpret mathematically the problem. I have to solve an equation of type $$\mathbb P\{X=k\}=0.1$$ but I can't identify the parameters.

1 Answers 1

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They are normal random variable, there is still a non-nil probability that the any machine will have a life span smaller than 160 days.

I guess you must calculate the expectation of the number of machines working after 160 days. Using your notation, we calculate $E(S_{200})$.

Given that the variables are iid, $$E(S_{200})=200E(X_1)=200E(1*1_{\{X_1=1\}}+0*1_{\{X_1=0\}})=200P(X_1=1)$$

Let $Z \sim \mathcal N(174,29^2)$, and $Y$ a standard normal variable we have

$$P(X_1=1)=P(Z>160)=P(174+29Y>160)=P(Y>-\frac{14}{29})$$

I let you conclude.

For the second part : You have to find m such as $$P(Z>m)<0.1$$, in other words, solve

$$P(174+29Y>m)=0.1$$

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    Thank you for your answer. In the first part, what is the law of $X_i$ ? I mean, what is $E(X_1)$ ? – 2017-02-09
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    You constructed $X_i$ , by construction, it is a Bernouilli distribution of parameter $p=P(Z>160)$ – 2017-02-09
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    with $Z\sim \mathcal N(174, 29^2)$ or $\mathcal N(0,1)$ ? – 2017-02-09
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    $Z\sim \mathcal N(174, 29^2)$ . $X_1$ equals one if the i-th machine works after 160 days, 0 otherwise. It is a binary random variable, then you ask yourself, what is the parameter of this binary variable? In other words, what is the probability that the i-th machine works after 160 days? You read your assumption and you see that a machine is governed by the distribution $Z\sim \mathcal N(174, 29^2)$ , hence my parameter is $P(Z>160)$ – 2017-02-09