Question

Four coins are tossed.Let Z be the random variable representing of heads that occur.Construct a table and find the values of random variable Z.

1.Determine the sample space.Let represent the tail and it represent the head.

2.Count the number of head in each outcome in the sample space and assign this outcome.

3.Indicate the summary of the possible values of random variable.

Answer 1

1. Let's denote $H$ - heads, $T$ - tails.

There are $2^4=16$ possible outcomes

$$S=\{HHHH, HHHT, HHTH, HTHH, THHH,$$

$$HHTT, HTHT, HTTH, THTH, THHT, TTHH,$$

$$HTTT, THTT, TTHT, TTTH, TTTT\}$$

2. The possible values of the random variable $Z$ are $0, 1, 2, 3, 4.$

$$\def\arraystretch{1.5} \begin{array}{c:c} Possible \ Outcomes & Z \\ \hline HHHH & 4 \\ \hdashline HHHT & 3 \\ \hdashline HHTH & 3 \\ \hdashline HTHH & 3 \\ \hdashline THHH & 3 \\ \hdashline HHTT & 2 \\ \hdashline HTHT & 2 \\ \hdashline HTTH & 2 \\ \hdashline THTH & 2 \\ \hdashline THHT & 2 \\ \hdashline TTHH & 2 \\ \hdashline HTTT & 1 \\ \hdashline THTT & 1 \\ \hdashline TTHT & 1 \\ \hdashline TTTH & 1 \\ \hdashline TTTT & 0 \\ \hdashline \end{array}$$

3. We will assume that the probability of getting heads and tails is the same:

$$p = q =1/2$$ $$\def\arraystretch{1.5} \begin{array}{c:c} Possible \ Outcomes & Z \\ \hline HHHH & 4 \\ \hdashline HHHT & 3 \\ \hdashline HHTH & 3 \\ \hdashline HTHH & 3 \\ \hdashline THHH & 3 \\ \hdashline HHTT & 2 \\ \hdashline HTHT & 2 \\ \hdashline HTTH & 2 \\ \hdashline THTH & 2 \\ \hdashline THHT & 2 \\ \hdashline TTHH & 2 \\ \hdashline HTTT & 1 \\ \hdashline THTT & 1 \\ \hdashline TTHT & 1 \\ \hdashline TTTH & 1 \\ \hdashline TTTT & 0 \\ \hdashline \end{array}$$

Construct the probability distribution of the random variable

$$\def\arraystretch{1.5} \begin{array}{c:c} z & 0 & 1 & 2 & 3 & 4 \\ \hline p(z) & 1/16 & 1/4 & 3/8 & 1/4 & 1/16 \end{array}$$

or

$$\def\arraystretch{1.5} \begin{array}{c:c} z & 0 & 1 & 2 & 3 & 4 \\ \hline p(z) & 0.0625 & 0.25 & 0.375 & 0.25 & 0.0625 \end{array}$$