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I have a distribution that is known to satisfy:

1) 30% of the distribution falls above $77,000$.

2) 20% of the distribution falls above $87,000$.

3) The distribution is normal

How can I infer the mean and standard deviation of the distribution?

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    Understood. Am sorry. Am trying. I will not use it as an excuse again. And thanks. Appreciate. – 2017-02-05
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    Glad to hear, Bondon. I really do welcome you here! It gets easier, once you get the hang of the site. I've deleted my comment, because I trust your resolve. – 2017-02-05
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    Thanks! It's really comforting to know that people on the internet are still nice! Will work hard! – 2017-02-05

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The golden rule of this site is that people are willing to help you with your question if you show that you have tried already. One way to do this is to show what you've got and where you are stuck. What you've got? Where are you stuck.

Hint: Normal distribution is fully determined by mean $\mu$ and standard deviation $\sigma$. Then your data write $$\begin{aligned} 1-\Phi_{\mu,\sigma}(77000)=0.3\\ 1-\Phi_{\mu,\sigma}(87000)=0.2\\ \end{aligned}$$ where $\Phi_{\mu,\sigma}$ is cdf of normal distribution with mean $\mu$ and standard deviation $\sigma$.