A question with probablity mass functions, cumulative distribution functions, mean, and st. dev.?

The question is as follows (I tried it on my own and had no idea where to start--our teacher taught us how to do it if the die is rolled once, but not 5 times):





Suppose you roll a die 5 times. Let X be the number of sixes that were rolled.


a) Write down the probability mass function (pmf) for X for all the values that X can take.


b) Write down the cumulative distribution function (cdf) for X.


c) Calculate the mean of X


d) Calculate the standard deviation of X.





Please break down how you got to the answer so that I can learn it...I learn best from examples and unfortunately our teacher didn't give us any.

A question with probablity mass functions, cumulative distribution functions, mean, and st. dev.?
Let X be the number of 6's in five rolls.





X has the binomial distribution with n = 5 trails and success probability p = 1/6 (for a fair die)





In general, if X has the binomial distribution with n trials and a success probability of p then


P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)


for values of x = 0, 1, 2, ..., n


P[X = x] = 0 for any other value of x.





this is found by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.





the mean of the binomial distribution is n * p


the variance of the binomial distribution is n * p * (1 - p)





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the CDF is:





P(X ≤ x) =





x


ΣP(X = i)


i = 0





P(X ≤ 0) = P(X = 0) = 0.4018776


P(X ≤ 1) = P(X = 0) + P(X = 1) = 0.8037551


P(X ≤ 2) = 0.9645062


P(X ≤ 3) = 0.9966564


P(X ≤ 4) = 0.9998714


P(X ≤ 5) = 1





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the mean is n*p = 5 * 1/6 = 5/6





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the standard deviation is the square root of the variance





variance = n* p * (1-p)


= 5 * 1/6 * 5/6 = 0.694444444





standard deviation is sqrt(5 * 1/6 * 5/6) = 0.8333333
Reply:a) You should have some equation in your book that handles the sum of two pmfs. The pmf for one roll of the douse (never say die!) is a uniform distribution over the integers [1,6]. Add two of those together ... then two more ... and one last roll, to total five. Or, if your equation is amenable to scaling, just apply the proper multiple of 5 in a critical spot (at this level of stats, my formulae didn't cover that).





b) Given that, the cdf is merely the sumulative sum of the pmf values.





c) %26amp; d) These formulae should also be in your text. The mean of the sum of distributions is the sum of the means. So, find the mean for one roll, and multiply by 5.


Similarly, the variance (standard deviation squared) of the sum of distributions is the sum of the variances. Find the variance for one roll, multiply by 5, and take the square root.