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Let Y1 and Y2 be independent exponentially distributed random variables, each with mean 7. Find P(Y1 > Y2 | Y1 < 2Y2). (En

ArbitrLikvidat [17]

Y₁ and Y₂ are independent, so their joint density is

$f_{Y_1,Y_2}(y_1,y_2)=f_{Y_1}(y_1)f_{Y_2}(y_2)=\begin{cases}\frac1{49}e^{-\frac{y_1+y_2}7}&\text{for }y_1\ge0,y_2\ge0\\0&\text{otherwise}\end{cases}$

By definition of conditional probability,

P(Y₁ > Y₂ | Y₁ < 2 Y₂) = P((Y₁ > Y₂) and (Y₁ < 2 Y₂)) / P(Y₁ < 2 Y₂)

Use the joint density to compute the component probabilities:

• numerator:

$P((Y_1>Y_2)\text{ and }(Y_1$

$=\displaystyle\frac1{49}\int_0^\infty\int_{\frac{y_1}2}^{y_1}e^{-\frac{y_1+y_2}7}\,\mathrm dy_2\,\mathrm dy_1$

$=\displaystyle-\frac17\int_0^\infty\int_{-\frac{3y_1}{14}}^{-\frac{2y_1}7}e^u\,\mathrm du\,\mathrm dy_1$

$=\displaystyle-\frac17\int_0^\infty\left(e^{-\frac{2y_1}7} - e^{-\frac{3y_1}{14}}\right)\,\mathrm dy_1$

$=\displaystyle-\frac17\left(-\frac72e^{-\frac{2y_1}7} + \frac{14}3 e^{-\frac{3y_1}{14}}\right)\bigg|_0^\infty$

$=\displaystyle-\frac17\left(\frac72 - \frac{14}3\right)=\frac16$

• denominator:

$P(Y_1$

(I leave the details of the second integral to you)

Then you should end up with

P(Y₁ > Y₂ | Y₁ < 2 Y₂) = (1/6) / (2/3) = 1/4

5 0

2 years ago

Please help thanks!

babunello [35]

Answer:

10 to 15

Step-by-step explanation:

the spread is the start and stop of points data. It goes from the least point to the most point, here there is a dot (two technally) on 10 and a dot on 15 where the data stops so the data spread is 10-15

7 0

3 years ago

Which equation represents a nonlinear function?

Grace [21]

Answer:

B and C

Step-by-step explanation:

Linear functions are written in the format y=mx+b

If a function has an exponent it is nonlinear

6 0

3 years ago

The collection of whole numbers is an example of a set

aev [14]

Answer:What are Whole Number? ... 17, 99, 267, 8107 and 999999999 are examples of whole numbers. Whole numbers include natural numbers that begin from 1 onwards. Whole numbers include positive integers along with 0.

Step-by-step explanation:

6 0

3 years ago

Towns X, Y, and Z are on the same straight road. Town Y is between Town X and Town Z. Town X is 57 miles from Town Z, and Town Y

Ray Of Light [21]

The distance between town X and Town Y is equal to the difference between the distance between Town X and Town Z and the distance from Town Y and Town Z. Mathematically
d (X-Y) = d (X-Z) - d(Y-Z)
Substituting,
d(X-Y) = 57 miles - 28 miles = 29 miles
Thus, the distance from Town X to Town Y is 29 miles.

6 0

3 years ago

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