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According to august 2011 survey, 40% of the adult cell phone owners downloaded an application to their cell phone.

Above table gives the probability distribution of X

From the table given above, we can calculate following values, that is:

$$\begin{gathered} \stackrel{}{鈭�\mathrm{P}}\left(\mathrm{x}\right)=1 \\ \stackrel{}{鈭�\mathrm{X}脳\mathrm{P}}\left(\mathrm{X}\right)=4.655 \\ \stackrel{}{鈭�{\mathrm{X}}^2脳\mathrm{P}}\left(\mathrm{X}\right)=41.525 \end{gathered}$$

We know that the requirements for a discrete probability distribution are

1. $0鈮�\mathrm{P}\left(\mathrm{x}\right)鈮�1$

2.$\underset{all\mathit{}y}{鈭�}\mathrm{P}\left(\mathrm{x}\right)=1$

Therefore, from the given data,

We can see that,

$0鈮�\mathrm{P}\left(\mathrm{x}\right)鈮�1$

And also,$\stackrel{}{鈭�\mathrm{P}}\left(\mathrm{x}\right)=1$

Hence the probability distribution is satisfying the given conditions for a discrete random variable.

Therefore$\mathrm{P}(\mathrm{X}鈮�10)=0.14$

c.

c.

Let the mean of X is given by:

$$\begin{gathered} \mathrm{E}\left(\mathrm{X}\right)=\underset{}{鈭�}\mathrm{x}脳\mathrm{P}\left(\mathrm{x}\right) \\ =4.655 \end{gathered}$$

Therefore,

$E\left(X\right)=4.655$

The variance is given by:

$\mathit{V}\mathit{a}\mathit{r}\mathbf{\left(}\mathit{X}\mathbf{\right)}\mathbf{=}\mathit{E}\mathbf{\left(}{\mathit{X}}^{\mathbf{2}}\mathbf{\right)}\mathbf{-}{\left[E\left(X\right)\right]}^{\mathbf{2}}$

$$\begin{gathered} \mathrm{Var}\left(\mathrm{X}\right)=\underset{}{鈭�}{\mathrm{x}}^2脳\mathrm{P}\left(\mathrm{x}\right)-{\left[\underset{}{鈭�}\mathrm{x}脳\mathrm{P}\left(\mathrm{x}\right)\right]}^2 \\ =41.525-21.669025 \\ =19.855975 \\ \mathrm{Var}\left(\mathrm{X}\right)=19.855975 \end{gathered}$$

d.

d.

To calculate an interval that will contain the value of x with a probability of at least .75, we specify

$\begin{array}{l}1\frac{1}{k^2}=0.75\\ \frac{1}{k^2}=1-0.75\\ \frac{1}{k^2}=0.25\\ k^2=\frac{1}{0.25}\\ =4\\ =2\end{array}$

Thus, the interval$渭-2蟽$ to$渭+2蟽$ will contain at least 75% of the probability.

This interval is given by

$$\begin{gathered} 渭-2蟽=4.655-2\left(4.56004\right) \\ =4.655-8.912008752 \\ =-4.25700875 \end{gathered}$$

Hence,$\mathrm{渭}-2\mathrm{蟽}=-4.26$

$$\begin{gathered} 渭-2蟽=4.655+2\left(4.56004\right) \\ =4.655-8.91200 \\ =13.5670 \end{gathered}$$

Hence,$渭-2蟽=13.56$ .