Question

Let X denote the temperature (degree C) and let Y denote thetime in minutes that it takes for the diesel engine on anautomobile to get ready to start. Assume that the joint density for(X,Y) is given by fxy (x,y) = c(4x + 2y +1) ; 0 < x< 40 and 0 < y <2.
(a) Find the value of c that makes this a density.
(b) Find the probability that on a randomly selected day theair temperature will exceed 20 degrees C and it will take at least1 minute for the car to be ready to start.
(c) Find the marginal densities for X and Y.
(d) Find the probability that on a randomly selected day itwill take at least one minute for the car to be ready tostart.
(e) Find the probability that on a randomly selected day theair temperature will exceed 20 degrees C.
(f) Are X and Y independent? Explain on mathematicallybasis.

Answer 1

Answer:

Step-by-step explanation:

Given $f_{XY} (x,y) = c(4x + 2y +1) ; 0 < x < 40\,and\, 0 < y$

a)

we know that $\int\limits^\infty_{-\infty}\int\limits^\infty_{-\infty} {f(x,y)} \, dxdy=1$

therefore $\int\limits^{40}_{-0}\int\limits^2_{0} {c(4x+2y+1)} \, dxdy=1$

on integrating we get

c=(1/6640)

b)

$P(X>20, Y>=1)=\int\limits^{40}_{20}\int\limits^2_{1} {\frca{1}{6640}(4x+2y+1)} \, dxdy$

on doing the integration we get

                        =0.37349

c)

marginal density of X is

$f(x)=\int\limits^2_{0} {\frca{1}{6640}(4x+2y+1)} \, dy$

on doing integration we get

f(x)=(4x+3)/3320 ; 0<x<40

marginal density of Y is

$f(y)=\int\limits^{40}_{0} {\frca{1}{6640}(4x+2y+1)} \, dx$

on doing integration we get

$f(y)=\frac{(y+40.5)}{83}$

d)

$P(01)=\int\limits^{40}_{0}\int\limits^2_{1} {\frca{1}{6640}(4x+2y+1)} \, dxdy$

solve the above integration we get the answer

e)

$P(X>20, 0$

solve the above integration we get the answer

f)

Two variables are said to be independent if there jointprobability density function is equal to the product of theirmarginal density functions.

we know f(x,y)

In the (c) bit we got f(x) and f(y)

f(x,y)cramster-equation-2006112927536330036287f(x).f(y)

therefore X and Y are not independent