Question

2. Suppose we toss four pennies. Let X be the total number of heads showing on ali the pennies Find the probability distribution of x. 3. Suppose we roll one four-sided die and one six-sided die. Let X be the sum of the pips showing on the two dice. Find the probability distribution for x. 4. The density curve of X is the triangle shown below: a) What is the height of the triangle. b) What is the probability that X is less than 1? Sketch the density curve, shade the area that represents the probability then fin the area. Do this for (c) also.) c) What is the probability that X is less than 0.5?

Answer 1

2) Let's assume that P(Head) = 1/2 and P(Tail) = 1/2

Here n = 4

X = number of heads in 4 tosses.

Therefore, X takes values as 0, 1, 2, 3, 4.

Therefore X follows binomial distribution with parameters n=4 and p = 1/2 = 0.5

Let's use excel:

The formulae used on the above excel-sheet are as follows:

p(x) BINOMDIST(A2,4,0.5,0) BINOMDIST(A3,4,0.5,0) BINOMDIST(A4,4,0.5,0) BINOMDIST(A5,4,0.5,0) BINOMDIST(A6,4,0.5,0)

3) Let X = sum of the pips showing on the two dice.

here total outcomes = 4 * 6 = 24

X takes values from 2 to 10

P( X = 2) = P( D1 = 1, D2 = 1) = (1/4)*(1/6) = 1/24

Here D1 = four sided die and D2 = six sided die

P( X = 3) = P( D1 = 1, D2 = 2) + P(D1 = 2, D2 = 1) = (1/24) + (1/24) = 2/24

P(X = 4) = P( D1 = 1, D2 = 3) + P(D1 = 2, D2 = 2) + P(D1 = 3, D2 = 1) = 3/24

P(X = 5) = P( D1 = 1, D2 = 4) + P(D1 = 2, D2 = 3) + P(D1 = 3, D2 = 2) + P(D1 = 4, D2 = 1) = 4/24

P(X = 6) = P( D1 = 1, D2 = 5) + P(D1 = 2, D2 = 4) + P(D1 = 3, D2 = 3) + P(D1 = 4, D2 = 2) = 4/24

P(X = 7) = P( D1 = 1, D2 = 6) + P(D1 = 2, D2 = 5) + P(D1 = 3, D2 = 4) + P(D1 = 4, D2 = 3) = 4/24

P(X = 8) = P(D1 = 2, D2 = 6) + P(D1 = 3, D2 = 5) + P(D1 = 4, D2 = 4) = 3/24

P(X = 9) = P(D1 = 3, D2 = 6) + P(D1 = 4, D2 = 5) = 2/24

P(X = 9) = P(D1 = 4, D2 = 6) = 1/24

Let's make table:

X	2	3	4	5	6	7	8	9	10
p(x)	1/24	2/24	3/24	4/24	4/24	4/24	3/24	2/24	1/24