Question

Supposefour coins are tossed. Let X represents the number of tails that occur. Illustrate a probability distribution of a random variable X USING FRACTION

Answer 1

There are $2^4=16$ outcomes.

$$HHHH, HHHT, HHTH, HTHH,$$

$$HHTT, HTHT, HTTH,HTTT,$$

$$TTTT, TTTH, TTHT, THTT,$$

$$TTHH, THTH, THHT, THHH$$

Let $X$ represents the number of tails that occur.

$$\begin{matrix} x & 0 & 1 & & 3 & 4 \\ \\ p(x) & \dfrac{1}{16} & \dfrac{4}{16} & \dfrac{6}{16} & \dfrac{4}{16} & \dfrac{1}{16} \end{matrix}$$

$$mean=E(X)=\sum_ix_ip(x_i)$$

$$=0(\dfrac{1}{16})+1(\dfrac{4}{16})+2(\dfrac{6}{16})+3(\dfrac{4}{16})+4(\dfrac{1}{16})=2$$

$$E(X^2)=\sum_ix_i^2p(x_i)$$

$$=0^2(\dfrac{1}{16})+1^2(\dfrac{4}{16})+2^2(\dfrac{6}{16})+3^2(\dfrac{4}{16})+4^2(\dfrac{1}{16})=5$$

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2=5-(2)^2=1$$

$$\sigma=\sqrt{\sigma^2}=1$$