Given the “heads” probability of a coin, I calculated the probability distribution of the random variable ...

Given the “heads” probability of a coin, I calculated the probability distribution of the random variable X ≡ number of heads in 5 independent tosses of this coin. This probability distribution is given below: (See attached file)

(a) Calculate the expectation of this random variable, i.e., E(X). Show your calculations for full credit. (5 pts.)

(b) Using your answer to part (a), infer the “heads” probability of this coin. Show your work for full credit. (5 pts.)

(c) I have a deal with my friend whereby this coin will be tossed 10 times and I will pay him a dollar amount equal to the square of the count of “heads” in these ten coin- tosses. (The other part of the deal is that he will give me a $15 gadget no matter what.) Tell me if this is a good deal for me, by calculating how much I should expect to pay my friend. Show your calculation for full credit. Hint: Variance of any random variable X is = E(X2) – (E(X))2. Also, we know a simple formula for variance of the relevant probability distribution. (10 points)

(d) You are analyzing a DNA sequence of length 1027 nucleotides, and you see the pattern ACGA 10 times in this sequence. The probability of seeing this pattern at any random position is 1/256. Calculate a “z-score” for your observation. The z-score of a random variable X is defined as: Z = (X – E(X))/sqrt(variance(X)). It is a common measure of how surprising an observation is. Show your calculations for full credit. Hint: assume that the occurrence of the pattern at any position is independent from its occurrence at any other position. (5 points)