# Homework 3 - Aidan Sharpe
## 1
$$F(x) = \begin{cases}
0 & x < 0 \\
1 - e^{-7x} & x \ge 0
\end{cases}$$

### a)
Find the probability density function of $X$:
$$f(x) = {d\over dx} F(x) = \begin{cases}
0 & x < 0 \\
7e^{-7x} & x \ge 0
\end{cases}
$$

### b)
Verify that $f(x)$ is a valid density function:
$$\int\limits_0^\infty f(x)dx \overset{?}{=} 1$$
$$\int\limits_0^\infty 7e^{-7x}dx = \left[-e^{-7x}\Big|_0^\infty\right] = 0 - (-1) = 1$$

### c)
Find $P(X<0.25)$:

$$F(0.25) = 1-e^{-7(0.25)} = 0.826$$

### d)
Find $P({5\over60} < X < {11\over60})$:
$$F\left({11\over60}\right) = 0.723$$
$$F\left({5\over60}\right) = 0.442$$
$$P\left({5\over60} < X < {11\over60}\right) = 0.723 - 0.442 = 0.281$$

## 2
$$f(x) = \begin{cases}
kx^{-3} & x > 1 \\
0 & \text{elsewhere}
\end{cases}$$

### a)
Find $k$, such that $f(x)$ is a valid density function:
$$\int\limits_0^\infty f(x)dx \overset{!}{=} 1$$
$$\int\limits_1^\infty kx^{-3}dx = \left[{-k\over 2x^2}\Big|_1^\infty\right] = 0 - {-k\over2}$$
$$\therefore k = 2$$

### b)
Find $F(x)$:
$$F(x) = \int\limits_{-\infty}^x f(t)dt$$
$$F(x) = \begin{cases}
1-{1\over x^2} & x > 1 \\
0 & \text{elsewhere}
\end{cases}$$

### c)
Find $P(X>7)$:
$$P(X>7) = F(7) = 0.98$$

### d)
Find $P(5 < X < 12)$:
$$P(X < 12) = F(12) = 0.993$$
$$P(X < 5) = 0.96$$
$$P(5 < X < 12) = F(12) - F(5) = 0.033$$

## 3
$$f(x) = \begin{cases}
{1\over5} & 1 < x < 6 \\
0 & \text{elsewhere}
\end{cases}$$

### a)
Verify that $f(x)$ is a valid probability density function:
$$\int\limits_0^\infty f(x)dx \overset{?}{=} 1$$
$${1\over5}(6 - 1) = 1$$

### b)
Find $P(2.5 \le X < 3)$:
$$P(2.5 \le X < 3) = F(3) - F(2.5) = 0.4 - 0.3 = 0.1$$

### c)
Find $P(X \le 2)$:
$$P(X \le 2) = F(2) = 0.4$$

### d)
Find $F(x)$:
$$F(x) = \int\limits_{-\infty}^{x} f(t)dt$$
$$F(x) = \begin{cases}
0 & x < 1 \\
{(x-1)\over5} & 1 \le x < 6 \\
1 & x \ge 6
\end{cases}$$

## 4
A box contains 7 dimes and 5 nickels. Three coins are chosen. $T$ is their total value in cents.

### a)
The set of all possible drawings of coins is:
$$\{DDD, DDN, DND, DNN, NDD, NDN, NND, NNN\}$$

Therefore the following are values for $T$:
$$T = \{30, 25, 20, 15\}$$

### b)
Find the probability density function for $T$, $f(x)$:
$$P(T = 30) = {{7\choose3}{5\choose0}\over{12\choose3}} = 0.159$$
$$P(T = 25) = {{7\choose2}{5\choose1}\over{12\choose3}} = 0.477$$
$$P(T = 20) = {{7\choose1}{5\choose2}\over{12\choose3}} = 0.318$$
$$P(T = 15) = {{7\choose0}{5\choose3}\over{12\choose3}} = 0.046$$

$$f(x) = \begin{cases}
0.046 & x = 15 \\
0.318 & x = 20 \\
0.477 & x = 25 \\
0.159 & x = 30 \\
\end{cases}$$

### c)
Find the CDF for $T$, $F(x)$:
$$f(x) = \begin{cases}
0 & x < 15 \\
0.046 & 15 \le x < 20 \\
0.364 & 20 \le x < 25\\
0.841 & 25 \le x < 30 \\
1 & x \ge 30 \\
\end{cases}$$

## 5
|$x$|$f(x)$|
|-|-
|10|0.08
|11|0.15
|12|0.30
|13|0.20
|14|0.20
|15|0.07

Determine the mean number of messages sent per hour
$$\sum\limits_x xf(x) = 10(0.08) + 11(0.15) + 12(0.30) + 13(0.20) + 14(0.20) + 15(0.07) = 12.5$$

## 6
Find the expected value for each of the following probability density functions:

### a)
$$f(x) = \begin{cases}
x + {1\over2} & 0 < x < 1 \\
0 & \text{elsewhere}
\end{cases}$$

$$E[x]=\int\limits_0^1 \left(x+{1\over2}\right)dx = \left[ {x^2 + x \over 2} \Big|_0^1 \right] =1$$

### b)
$$f(x) = \begin{cases}
{3\over x^4} & x \ge 1 \\
0 & \text{elsewhere}
\end{cases}$$

$$E[x] \int\limits_1^\infty {3\over x^4}dx = \left[ {-1 \over x^3} \Big|_1^\infty \right] = 0 - (-1) = 1$$