91Ó°ÊÓ

Probability denotes the possibility of something happening. It is a field of mathematics that studies the probability of a random event occurring.From\({\rm{0}}\)to\({\rm{1}}\), the value is expressed. Probability is a mathematical concept that predicts how likely occurrences are to occur.

Let \({\rm{X}}\)and \(Y\)be continuous random variables.

The joint probability density function

\({\rm{f(x,y)}}\)is a function that is non-negative and for which

\(\int_{ - \infty }^\infty {\int_{ - \infty }^\infty f } {\rm{(x,y)dxdy = 1}}\)

For every adequate set \({\rm{A}}\)the following holds

(a):

From relation

\(\int_{ - \infty }^\infty {\int_{ - \infty }^\infty f } {\rm{(x,y)dxdy = 1}}\)

we can find value of \({\rm{K}}\).

The area \(20 \le x \le {\rm{30,20}} \le {\rm{y}} \le 30\) is a simple square, therefore the following holds

\(\begin{array}{l}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty f } {\rm{(x,y)dxdy = }}\int_{{\rm{20}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{30}}} {\rm{K}} } \left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right){\rm{dxdy = K}}\int_{{\rm{20}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{2}}}} } {\rm{dydx + K}}\int_{{\rm{20}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{y}}^{\rm{2}}}} } {\rm{dxdy}}\\{\rm{ = K}}\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{2}}}} \left( {\left. {\rm{y}} \right|_{{\rm{20}}}^{{\rm{30}}}} \right){\rm{dx + K}}\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{y}}^{\rm{2}}}} \left( {\left. {\rm{x}} \right|_{{\rm{20}}}^{{\rm{30}}}} \right){\rm{dy}}\\{\rm{ = }}\left. {{\rm{10K \times }}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{30}}}{\rm{ + }}\left. {{\rm{10K \times }}\frac{{{{\rm{y}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{30}}}\\{\rm{ = 2 \times 10K}}\left( {\frac{{{\rm{3}}{{\rm{0}}^{\rm{3}}}}}{{\rm{3}}}{\rm{ - }}\frac{{{\rm{2}}{{\rm{0}}^{\rm{3}}}}}{{\rm{3}}}} \right)\\{\rm{ = 20K \times }}\frac{{{\rm{19,000}}}}{{\rm{3}}}\\{\rm{ = }}\frac{{{\rm{380,000}}}}{{\rm{3}}}{\rm{ \times K}}\end{array}\)

Hence, we have

\(\frac{{{\rm{380,000}}}}{{\rm{3}}}{\rm{ \times K = 1}}\)

or equally

Therefore,The value of \({\rm{K = }}\frac{{\rm{3}}}{{{\rm{380,000}}}}\)

(b):

The underfill limit is 26 psi, therefore we need the following probability

Therefore, The probability is \({\rm{P(X > 26}}\)and \({\rm{Y < 26) = 0}}{\rm{.3024;}}\)

(c):

We need to find subset of

\(20 \le x \le 30,\;\;\;20 \le y \le 30\)

for which the difference of \({\rm{X}}\)and \({\rm{Y}}\) is at most \({\rm{2}}\). We can represent the distance between two points \({\rm{x}}\) and \({\rm{y}}\)as \({\rm{|x - y|}}\).

Therefore, we need to find probability of event

\(\{ |{\rm{X - Y|}} \le 2\} \)

Look at the picture. We need to integrate over region \({\rm{II}}\). The following holds

We have that

\(\int_{{\rm{20}}}^{{\rm{28}}} {\int_{{\rm{x + 2}}}^{{\rm{30}}} {\rm{f}} } {\rm{(x,y)dydx = }}\int_{{\rm{22}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{x - 2}}} {\rm{f}} } {\rm{(x,y)dydx}}\)

however, see the following algebra behind the integrals:

\(\begin{array}{l}{{\rm{I}}_{\rm{1}}}{\rm{ = }}\int_{{\rm{20}}}^{{\rm{28}}} {\int_{{\rm{x + 2}}}^{{\rm{30}}} {\rm{K}} } \left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right){\rm{dydx}}\\{\rm{ = K}}\int_{{\rm{20}}}^{{\rm{28}}} {\int_{{\rm{x + 2}}}^{{\rm{30}}} {\left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right)} } {\rm{dydx}}\\{\rm{ = K}}\int_{{\rm{20}}}^{{\rm{28}}} {\int_{{\rm{x + 2}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{2}}}} } {\rm{dydx + K}}\int_{{\rm{20}}}^{{\rm{28}}} {\int_{{\rm{x + 2}}}^{{\rm{30}}} {{{\rm{y}}^{\rm{2}}}} } {\rm{dydx}}\\{\rm{ = K}}\int_{{\rm{28}}}^{{\rm{28}}} {{{\rm{x}}^{\rm{2}}}\left( {\left. {\rm{y}} \right|_{{\rm{x + 2}}}^{{\rm{30}}}} \right)} {\rm{dx + K}}\int_{{\rm{20}}}^{{\rm{28}}} {\left( {\left. {\frac{{{{\rm{y}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{x + 2}}}^{{\rm{30}}}} \right)} {\rm{dx}}\\{\rm{ = K}}\underbrace {\int_{{\rm{20}}}^{{\rm{28}}} {{{\rm{x}}^{\rm{2}}}} {\rm{(30 - x - 2)dx}}}_{{{\rm{I}}_{\rm{3}}}}{\rm{ + }}\frac{{\rm{K}}}{{\rm{3}}}\underbrace {\int_{{\rm{20}}}^{{\rm{28}}} {\left( {{\rm{3}}{{\rm{0}}^{\rm{3}}}{\rm{ - (x + 2}}{{\rm{)}}^{\rm{3}}}} \right)} {\rm{dx}}}_{{{\rm{I}}_{\rm{4}}}}\end{array}\)

where we have

\(\begin{array}{l}{{\rm{I}}_{\rm{3}}}{\rm{ = 28}}\int_{{\rm{20}}}^{{\rm{28}}} {{{\rm{x}}^{\rm{2}}}} {\rm{dx - }}\int_{{\rm{20}}}^{{\rm{28}}} {{{\rm{x}}^{\rm{3}}}} {\rm{dx}}\\{\rm{ = }}\left. {{\rm{28 \times }}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{28}}}{\rm{ - }}\left. {\frac{{{{\rm{x}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{{\rm{20}}}^{{\rm{28}}}\\{\rm{ = 16,554}}{\rm{.67,}}\end{array}\)

similarly we get

\(\begin{array}{l}{{\rm{I}}_{\rm{4}}}{\rm{ = }}\int_{{\rm{20}}}^{{\rm{28}}} {\rm{3}} {{\rm{0}}^{\rm{3}}}{\rm{dx - }}\int_{{\rm{20}}}^{{\rm{28}}} {{{\rm{f}}_{\rm{1}}}} {\rm{(x)dx}}\\{\rm{ = 3}}{{\rm{0}}^{\rm{3}}}{\rm{ \times 8 - }}\left. {\frac{{{{\rm{x}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{{\rm{20}}}^{{\rm{28}}}{\rm{ - }}\left. {{\rm{6 \times }}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{28}}}{\rm{ - 12 \times }}\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}{\rm{ - }}\left. {{\rm{8 \times x}}} \right|_{{\rm{20}}}^{{\rm{28}}}\\{\rm{ = 72,064}}{\rm{.}}\end{array}\)

Where the function \({{\rm{f}}_{\rm{1}}}\) is

\(\begin{array}{l}{{\rm{f}}_{\rm{1}}}{\rm{(x) = (x + 2}}{{\rm{)}}^{\rm{3}}}\\{\rm{ = }}{{\rm{x}}^{\rm{3}}}{\rm{ + 3 \times }}{{\rm{x}}^{\rm{2}}}{\rm{ \times 2 + 3 \times x \times }}{{\rm{2}}^{\rm{2}}}{\rm{ + }}{{\rm{2}}^{\rm{3}}}\\{\rm{ = }}{{\rm{x}}^{\rm{3}}}{\rm{ + 6}}{{\rm{x}}^{\rm{2}}}{\rm{ + 12x + 8}}\end{array}\)

The first integral is

\(\begin{array}{l}{{\rm{I}}_{\rm{1}}}{\rm{ = K \times 16,554}}{\rm{.67 + }}\frac{{\rm{K}}}{{\rm{3}}}{\rm{ \times 72,064}}\\{\rm{ = 0}}{\rm{.3203}}\end{array}\)

For the second integral, we have

\(\begin{array}{l}{{\rm{I}}_{\rm{2}}}{\rm{ = }}\int_{{\rm{22}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{x - 2}}} {\rm{K}} } \left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right){\rm{dydx}}\\{\rm{ = K}}\int_{{\rm{22}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{x - 2}}} {\left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right)} } {\rm{dydx}}\\{\rm{ = K}}\int_{{\rm{22}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{x - 2}}} {{{\rm{x}}^{\rm{2}}}} } {\rm{dydx + K}}\int_{{\rm{22}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{x - 2}}} {{{\rm{y}}^{\rm{2}}}} } {\rm{dydx}}\\{\rm{ = K}}\int_{{\rm{20}}}^{{\rm{30}}} {\left( {\left. {\rm{y}} \right|_{{\rm{20}}}^{{\rm{x - 2}}}} \right)} {\rm{dx + K}}\int_{{\rm{22}}}^{{\rm{30}}} {\left( {\left. {\frac{{{{\rm{y}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{x - 2}}}} \right)} {\rm{dx}}\\{\rm{ = K}}\underbrace {\int_{{\rm{22}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{2}}}} {\rm{(x - 2 - 20)dx}}}_{{{\rm{I}}_{\rm{5}}}}{\rm{ + }}\frac{{\rm{K}}}{{\rm{3}}}\underbrace {\int_{{\rm{22}}}^{{\rm{30}}} {\left( {{{{\rm{(x - 2)}}}^{\rm{3}}}{\rm{ - 2}}{{\rm{0}}^{\rm{3}}}} \right)} {\rm{dx}}}_{{{\rm{I}}_{\rm{6}}}}\end{array}\)

where

\(\begin{array}{l}{{\rm{I}}_{\rm{5}}}{\rm{ = }}\int_{{\rm{22}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{3}}}} {\rm{dx - 22}}\int_{{\rm{22}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{2}}}} {\rm{dx}}\\{\rm{ = }}\left. {\frac{{{{\rm{x}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{{\rm{22}}}^{{\rm{30}}}{\rm{ - }}\left. {{\rm{22 \times }}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{22}}}^{{\rm{30}}}\\{\rm{ = 24,021}}{\rm{.33,}}\end{array}\)

and also

\(\begin{array}{l}{{\rm{I}}_{\rm{6}}}{\rm{ = }}\int_{{\rm{22}}}^{{\rm{30}}} {\underbrace {{{{\rm{(x - 2)}}}^{\rm{3}}}}_{{{\rm{f}}_{\rm{2}}}{\rm{(x)}}}} {\rm{dx - 2}}{{\rm{0}}^{\rm{3}}}{\rm{ \times }}\int_{{\rm{22}}}^{{\rm{30}}} {\rm{d}} {\rm{x}}\\{\rm{ = }}\left. {\frac{{{{\rm{x}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{{\rm{22}}}^{{\rm{30}}}{\rm{ - }}\left. {{\rm{6 \times }}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{22}}}^{{\rm{30}}}{\rm{ + }}\left. {{\rm{12 \times }}\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}} \right|_{{\rm{22}}}^{{\rm{30}}}{\rm{ - }}\left. {{\rm{8 \times x}}} \right|_{{\rm{22}}}^{{\rm{30}}}{\rm{ - }}\left. {{\rm{2}}{{\rm{0}}^{\rm{3}}}{\rm{ \times x}}} \right|_{{\rm{22}}}^{{\rm{30}}}\\{\rm{ = 49,664}}\end{array}\)

Where the function \({{\rm{f}}_{\rm{2}}}\)can be written as

\(\begin{array}{l}{{\rm{f}}_{\rm{2}}}{\rm{(x) = (x - 2}}{{\rm{)}}^{\rm{3}}}\\{\rm{ = }}{{\rm{x}}^{\rm{3}}}{\rm{ - 3 \times }}{{\rm{x}}^{\rm{2}}}{\rm{ \times 2 + 3 \times x \times }}{{\rm{2}}^{\rm{2}}}{\rm{ - }}{{\rm{2}}^{\rm{3}}}\\{\rm{ = }}{{\rm{x}}^{\rm{3}}}{\rm{ - 6 \times }}{{\rm{x}}^{\rm{2}}}{\rm{ + 12 \times x - 8}}\end{array}\)

The second integral is

\(\begin{array}{l}{{\rm{I}}_{\rm{2}}}{\rm{ = K \times 24,021}}{\rm{.33 + }}\frac{{\rm{K}}}{{\rm{3}}}{\rm{ \times 49,664}}\\{\rm{ = 0}}{\rm{.3203}}\end{array}\)

The probability is \({\rm{P(|X - Y|}} \le 2) = 0.3594\);

(d):

The continuous random variable's marginal probability mass function \({\rm{X}}\)is

\({{\rm{f}}_{\rm{X}}}{\rm{(x) = }}\int_{ - \infty }^\infty f {\rm{(x,y)dy,}}\;\;\;{\rm{ for }} - \infty < {\rm{x}} < \infty \)

The continuous random variable's marginal probability mass function \({\rm{Y}}\)is

\({{\rm{f}}_{\rm{Y}}}{\rm{(x) = }}\int_{ - \infty }^\infty f {\rm{(x,y)dx,}}\;\;\;{\rm{ for }} - \infty < y < \infty \)

The following holds

\(\begin{array}{l}{{\rm{f}}_{\rm{X}}}{\rm{(x) = }}\int_{{\rm{20}}}^{{\rm{30}}} {\rm{K}} \left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right){\rm{dy}}\\{\rm{ = 10K}}{{\rm{x}}^{\rm{2}}}{\rm{ + }}\left. {{\rm{K}}\frac{{{{\rm{y}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{30}}}\\{\rm{ = 10 \times K}}{{\rm{x}}^{\rm{2}}}{\rm{ + 0}}{\rm{.05,}}\\20 \le x \le 30,\\{f_X}(x) = 0,x \notin (20,30).\end{array}\)

The same marginal distribution would be obtained for \({\rm{Y}}\) if we substitute \({\rm{x}}\)with \({\rm{y}}\).

The distribution of air pressure is \({{\rm{f}}_{\rm{X}}}{\rm{(x) = }}\left\{ {\begin{array}{*{20}{l}}{10{\rm{ \times K}}{{\rm{x}}^{\rm{2}}}{\rm{ + 0}}{\rm{.05}}}&{,20 \le x \le 30}\\0&{,{\rm{ otherwise }}}\end{array}} \right.\)

(e):

Two independent variables If and only if, \({\rm{X}}\) and \({\rm{Y}}\) are independent.1. \({\rm{p(x,y) = }}{{\rm{p}}_{\rm{X}}}{\rm{(x) \times }}{{\rm{p}}_{\rm{Y}}}{\rm{(y)}}\), for every \({\rm{(x,y)}}\) and when \({\rm{X}}\)and \({\rm{Y}}\)discrete rv's,

2. \({\rm{f(x,y) = }}{{\rm{f}}_{\rm{X}}}{\rm{(x) \times }}{{\rm{f}}_{\rm{Y}}}{\rm{(y)}}\),

for every \({\rm{(x,y)}}\)and when \({\rm{X}}\)and \({\rm{Y}}\)continuous rv's,

otherwise they are dependent.

The random variables are obviously not independent. Because of this, functions

\(\begin{array}{c}{\rm{f(x,y) = K}}\left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right){{\rm{f}}_{\rm{X}}}{\rm{(x) \times }}{{\rm{f}}_{\rm{Y}}}{\rm{(y)}}\\{\rm{ = }}\left( {{\rm{10 \times K}}{{\rm{x}}^{\rm{2}}}{\rm{ + 0}}{\rm{.05}}} \right){\rm{ \times }}\left( {{\rm{10 \times K}}{{\rm{y}}^{\rm{2}}}{\rm{ + 0}}{\rm{.05}}} \right){\rm{,}}\end{array}\)

are not equal

Therefore,The random variables are not independent