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Into the Normal Distribution

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

three normal distributions
This chart accurately shows the probability density functions for three normal distributions.

I estimate that the probability that a variable drawn from the blue distribution is negative is $0.25$. Can you suggest how I made this estimate? Is it likely to be an over- or under-estimate?

How about the probabilities that random variables drawn from the other distributions are negative?

My friend says that the means and variances of these normal distributions are whole numbers. Can you use your estimates to work out the values for these means and variances, using just tables and estimates? Can you specify the values exactly, or can you only specify a certain range of possibilities?