4.3 KiB
Chapter 3
Example
Suppose there are 30 resistors, 7 of them do not work. You randomly choose 3 of them. Let X be the number of defective resistors. Find the probability distribution of X.
X = [0,3]
P(X=0) = { {7 \choose 0} {23 \choose 3} \over {30 \choose 3} } = 0.436
P(X=1) = { {7 \choose 1} {23 \choose 2} \over {30 \choose 3} } = 0.436
P(X=2) = { {7 \choose 2} {23 \choose 1} \over {30 \choose 3} } = 0.119
P(X=3) = { {7 \choose 3} {23 \choose 0} \over {30 \choose 3} } = 0.009
Probability distribution: $$P(X = x) = \begin{cases} 0.436 & x=0 \ 0.436 & x=1 \ 0.119 & x=2 \ 0.009 & x=3 \end{cases}
## The Cumulative Distribution Function
The cumulative distribution function (CDF), $F(x)$, of a discrete random variable, $x$, with probability distribution, $f(x)$, is:
$$F(x) = P(X \le x)$$
Find CDF for the example above:
$$F(0) = P(X \le 0) = P(X = 0) = 0.436$$
$$F(1) = P(X \le 1) = P((X = 0) \cup (X=1)) = 0.872$$
$$F(2) = P(X \le 2) = P((X=0) \cup (X=1) \cup (X=2)) = 0.991$$
Since 3 is the largest possible value for $x$:
$$F(3) = P(X \le 3) = 1$$
As a piecewise function:
$$F(x) =
\begin{cases}
0 & x < 0 \\
0.436 & 0 \le x < 1 \\
0.872 & 1 \le x < 2 \\
0.991 & 2 \le x < 3 \\
1 & x \ge 3
\end{cases}$$
### Exercise
Suppose that a days production of 850 manufactured parts contains 50 parts that to not conform to customer requirements. 2 parts are selected at random from the batch. Let $X$ be the number of non-conforming parts.
#### a)
Find the probability distribution for $X$
$$P(X = 0) = { {50 \choose 0} {800 \choose 2} \over {850 \choose 2 }} = 0.8857$$
$$P(X = 1) = { {50 \choose 1} {800 \choose 1} \over {850 \choose 2 }} = 0.1109$$
$$P(X = 2) = { {50 \choose 2} {800 \choose 0} \over {850 \choose 2 }} = 0.0034$$
$$P(X = x) =
\begin{cases}
0.8857 & x=0 \\
0.1109 & x=1 \\
0.0034 & x=2
\end{cases}
b)
Find the CDF F(x)
$$F(x) =
\begin{cases}
0 & x < 0 \
0.8857 & 0 \le x < 1 \
0.9966 & 1 \le x < 2 \
1 & x \ge 2
\end{cases}$$
c)
Plot F(x):
Continuous Probability Distributions
A continuous random variable is a variable that can take on any value within a range. It takes on infinitely many possible value within the range.
For a continuous distribution, f(x):
P(X = x) = 0
P(x_0 \le X \le x_1) = \int\limits_{x_0}^{x_1} f(x) dx
P(X \ge x_0) = \int\limits_{x_0}^{\infty} f(x) dx
Definition
The function, f(x), is a probability density function fo the continuous random variable, X, defined over \Reals if:
f(x) \ge 0, \forall x \in \Reals
\int\limits_{-\infty}^{\infty} f(x) dx = 1
P(x_0 \le X \le x_1) = P(x_0 < X < x_1)
= P(x_0 \le X < x_1)
= P(x_0 < X \le x_1)
Example
Suppose that the error in the reaction temperature in ^\circ \text{C} for a controlled lab experiment is a continuous random variable, X, having PDF:
$$f(x) =
\begin{cases}
{x^2 \over 3} & -1 < x < 2 \
0 & elsewhere
\end{cases}$$
a)
Verify that f(x) is a PDF.
\int\limits_{-1}^{2} {x^2 \over 3} dx \stackrel{?}{=} 1
{1 \over 3} \left[{1 \over 3} x^3 \Big\vert_{-1}^{2}\right] = {1\over9}[8- (-1)] = 1
b)
Find P(0 < X < 0.5):
P(0 < X < 0.5) = \int\limits_0^{0.5} {x^2 \over 3}dx
{1\over9}\left[x^3 \Big|_0^{0.5}\right] = {1\over9}[0.125] = 0.01389
Definition
The CDF, F(x) of a continuous random variabl, X, with probability density function f(x) is:
F(x) = P(X \le x) = \int\limits_{-\infty}^x f(t) dt
Note:
P(a < X < b) = F(b) - F(a)
f(x) = {d\over dx}F(x)
Example
Find the CDF of the previous example $$f(x) = \begin{cases} {x^2 \over 3} & -1 < x < 2 \ 0 & elsewhere \end{cases}$$
F(x) = \int\limits_{-1}^x {t^2 \over 3} dt
{1/over 9}\left[t^3\Big|_{-1}^x\right] = {1\over 9}\left[x^3 + 1\right]
$$F(x) = \begin{cases} 0 & t < -1 \ {1\over 9} \left[x^3 + 1\right] & -1 \le x \le 2 \ 1 & elsewhere \end{cases}$$
Example
The proportion of the budget for a certain type of industrial copany that is allotted to environmential and pollution control is coming under scrutiny. A data collection project determines that the distribution of these proportions is given by: $$f(y) = \begin{cases} k(1-y)^4 & 0 \le y \le 1 \ 0 & elsewhere \end{cases}$$
Find k that renders $f(y) a valid density function:
\int\limits_0^1 k(1-y)^4dy = 1
{k\over5} = 1
\therefore k = 5