Answer to Question #213453 in Statistics and Probability for dan

Question

8.A coin is loaded so that heads is three times as likely as the tails. For 3 independent tosses of the coin find the pmf of the total number of heads realised and the probability of realizing at most 2 heads.

9.Supposse that the random variable x has a p.d.f given by f(x)={cx,0<x<1 o ,elsewhere.Find the values of the constant c hence determine m. so that p(x<or equal to m)=1/2

Answer 1

8. Let $X=$ the number of heads realised: $X\sim Bin(n, p)$

Given $n=3, p=0.75, q=0.25$

$$P(X=x)=\dbinom{3}{x}(0.75)^x(0.25)^{3-x}, x=0,1,2,3$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 1/64 & 9/64 & 27/64 & 27/64 \\ \end{array}$$

$$P(X\leq2)=1-P(X=3)=1-\dfrac{27}{64}=\dfrac{37}{64}$$

$$=0.578125$$

9.

$$\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{0}^{1}cxdx$$

$$=c\big[\dfrac{x^2}{2}\big]\begin{matrix} 1 \\ 0 \end{matrix}=\dfrac{c}{2}=1=>c=2$$

$$\displaystyle\int_{0}^{m}2xdx=\dfrac{1}{2}, 0<m<1$$

$$[x^2]\begin{matrix} m \\ 0 \end{matrix}=\dfrac{1}{2}$$

$$m^2=\dfrac{1}{2}$$

Since $0<m<1,$ we take $m=\dfrac{\sqrt{2}}{2}.$