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If X takes the value 1 with probability p and 0 with probability (1-p), and there are n independent trials then E(X) = np
The binomial distribution.
A binomial is an algebraic expression. It does not have an area.
If this is the only information that you have then you must use the Poisson distribution.
This is a binomial distribution; number of trials (n) is 6, probability of success (p) is 1/2 or 0.5. With this information you can go to a Binomial Distribution Table and find the solution. Within the section of values for n=6 and p=.5, read from the section the probability of 2 which is 0.2344 (see related link for table).
The standard deviation is sqrt[n*p*(1-p)] = sqrt(1000*0.94*0.06) = 7.51 approx.
When the event of interest is a cumulative event. For example, to find the probability of getting three Heads in 8 tosses of a fair coin you would use the regular binomial distribution. But to find the probability of up to 3 Heads you would use the cumulative distribution. This is because Prob("up to 3") = Prob(0 or 1 or 2 or 3) = Prob(0) + Prob(1) + Prob(2) + Prob(3) since these are mutually exclusive.
Read the instructions that accompany the table: they do not all have exactly the same layout.
You seem to be referring to the Pearson chi-square test-of-fit statistic. To do this you need not only the observed values in a frequency table (which you have) but the expected (or theoretical) values for that table.In practical situations the expected values are obtained by making some educated guess about what distribution the observed values came from, estimating the parameters of that distribution and then using the estimated distribution to obtain the required expected values to calculate the chi-square.In short, you need more information.
Expected frequencies are used in a chi-squared "goodness-of-fit" test. there is a hypothesis that is being tested and, under that hypothesis, the random variable would have a certain distribution. The expected frequency for a "cell" is the number of observations that you would expect to find in that cell if the hypothesis were true.
how do i find the median of a continuous probability distribution
There is not enough information to find n & p. The mean is n*p and the std dev = sqrt (n*p*q). You have to be given n, p or q to have 2 equations 2 unknowns to solve.