Question

Consider a random experiment of throwing FOUR perfectly balanced and identical coins. Suppose that for each toss that comes up heads we win $4, but for each toss that comes up tails we lose $3. Clearly, a quantity of interest in this situation is our total wining. Let X denote this quantity. Answer the following questions.

(a) What are the values that the random variable X takes?

(b) Find P(X = 16) =? & P(X = 2) =? & P(X = −5) =?

(c) P(X ≥ 10) =? & P(X ≤ 9)=?

(d) (2 points) Find E(X), E(X2 ), and Var(X)

Answer 1

a)

Total number of outcomes = 24 = 16

Values that X can take:

X=4+4+4+4=16: {(HHHH)}

X=4+4+4-3=9: {(HHHT),(HHTH),(HTHH),(THHH)}

X=4+4-3-3=2: {(HHTT),(HTHT),(THHT),(TTHH),(HTHT),(HTTH)}

X=4-3-3-3=-5: {(HTTT),(THTT),(TTHT),(TTTH)}

X=-3-3-3-3=-12: {(TTTT)}

So, X can take values: 16,9,2,-5,-12

b)

Now,

c)

Required probabilities:

d)

Now,

So,