The Gaussian random variable $X$ can be used to model the number of customers that enter a market in 1 minute at a given time of the day. The mean number of customers that enter the market in 1 minute is 10 and the probability that fewer than 3 customers enter the market in 1 minute is $ 0.035 $. What is the probability that more than 12 customers enter the market in 1 minute. Also determine $f_X(x|X>12)$ ?
I couldn't find variance. So i couldn't solve this problem
Thanks in advance.
As @Seyhmus Güngören points out, you have been asked to solve a drill problem that is not likely to be useful in practice. So it may be difficult for you to have an intuitive sense how to proceed. Here is the approach I think you are expected to take.
$$0.035 = P(X \le 3) = P(Z \le (3 - 10)/\sigma),$$
where $Z$ is standard normal. From normal tables (or from software) you can find $P(Z \le -1.8119) = 0.035.$ From there you should be able to find the value of $\sigma.$
Notes: (1) Maybe you'd be asked to use $(3.5 - 10)/\sigma$ in the displayed equation. (2) The resulting normal distribution with mean 10 and $\sigma$ as computed above has almost half a percent of its area below 0. So this normal distribution is is hardly a realistic model for numbers of people entering the market. Not only that, the number of people is discrete and the normal distribution is continuous. A more realistic model would use the Poisson distribution, which you may study later.