Can someone walk me through the steps of answering these questions. I'm having a hard time understading probab

The Internal Revenue Service reports that 8% of individual tax returns in 2000 showed an adjusted gross income of $100,000 or more. (Hint: You can use the binomial probability distribution to answer these questions.)

--if you selected a random sample of 10 returns for 2000, what is the probability that more than 2 of them would show an adjusted gross income of $100,000 or more?

--If you completed the entire probability distribution for this problem, what shape would it have? Explain.

--What is the mean of this distribution. What does this number tell you?

Let X be the number of returns with a adjusted gross income of $100,000 or more. X has the binomial distribution with n = 10 trials and success probability p = 0.08

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.

The probability mass function is derived 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.
Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

X ~ Binomial( n , p )

the mean of the binomial distribution is n * p = 0.8
the variance of the binomial distribution is n * p * (1 - p) = 0.736
the standard deviation is the square root of the variance = 鈭?( n * p * (1 - p)) = 0.8579044

The Probability Mass Function, PMF,
f(X) = P(X = x) is:

P( X = 0 ) = 0.4343885
P( X = 1 ) = 0.3777291
P( X = 2 ) = 0.1478070
P( X = 3 ) = 0.03427410
P( X = 4 ) = 0.005215623
P( X = 5 ) = 0.0005442389
P( X = 6 ) = 3.943760e-05
P( X = 7 ) = 1.959633e-06
P( X = 8 ) = 6.390106e-08
P( X = 9 ) = 1.234803e-09
P( X = 10 ) = 1.073742e-11


The Cumulative Distribution Function, CDF,
F(X) = P(X 鈮?x) is:

x
鈭?P(X = t) =
t = 0

P( X 鈮?0 ) = 0.4343885
P( X 鈮?1 ) = 0.8121175
P( X 鈮?2 ) = 0.9599246
P( X 鈮?3 ) = 0.9941987
P( X 鈮?4 ) = 0.9994143
P( X 鈮?5 ) = 0.9999585
P( X 鈮?6 ) = 0.999998
P( X 鈮?7 ) 鈮? 1
P( X 鈮?8 ) 鈮? 1
P( X 鈮?9 ) 鈮? 1
P( X 鈮?10 ) = 1


1 - F(X) is:

n
鈭?P(X = t) =
t = x

P( X 鈮?0 ) = 1
P( X 鈮?1 ) = 0.5656115
P( X 鈮?2 ) = 0.1878825
P(X > 2) = P( X 鈮?3 ) = 0.04007542 鈫?answer to a
P( X 鈮?4 ) = 0.005801325
P( X 鈮?5 ) = 0.0005857013
P( X 鈮?6 ) = 4.146238e-05
P( X 鈮?7 ) = 2.024779e-06
P( X 鈮?8 ) = 6.51466e-08
P( X 鈮?9 ) = 1.245540e-09
P( X 鈮?10 ) = 1.07373e-11



if you drew out the distribution it would have a peak at x = 0 and then be skewed to the right.

the mean is 0.8. You expect less than one out of 10 returns to have a gross adjusted income greater than $100,000