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The probability density function of the time to failure of an electronic component in a copier (in hours) is f(x) for Determine

salantis [7]

The question is incomplete. Here is the complete question.

The probability density function of the time to failure of an electronic component in a copier (in hours) is

$f(x)=\frac{e^{\frac{-x}{1000} }}{1000}$

for x > 0. Determine the probability that

a. A component lasts more than 3000 hours before failure.

b. A componenet fails in the interval from 1000 to 2000 hours.

c. A component fails before 1000 hours.

d. Determine the number of hours at which 10% of all components have failed.

Answer: a. P(x>3000) = 0.5

b. P(1000<x<2000) = 0.2325

c. P(x<1000) = 0.6321

d. 105.4 hours

Step-by-step explanation: <em>Probability Density Function</em> is a function defining the probability of an outcome for a discrete random variable and is mathematically defined as the derivative of the distribution function.

So, probability function is given by:

P(a<x<b) = $\int\limits^b_a {P(x)} \, dx$

Then, for the electronic component, probability will be:

P(a<x<b) = $\int\limits^b_a {\frac{e^{\frac{-x}{1000} }}{1000} } \, dx$

P(a<x<b) = $\frac{1000}{1000}.e^{\frac{-x}{1000} }$

P(a<x<b) = $e^{\frac{-b}{1000} }-e^\frac{-a}{1000}$

a. For a component to last more than 3000 hours:

P(3000<x<∞) = $e^{\frac{-3000}{1000} }-e^\frac{-a}{1000}$

Exponential equation to the infinity tends to zero, so:

P(3000<x<∞) = $e^{-3}$

P(3000<x<∞) = 0.05

There is a probability of 5% of a component to last more than 3000 hours.

b. Probability between 1000 and 2000 hours:

P(1000<x<2000) = $e^{\frac{-2000}{1000} }-e^\frac{-1000}{1000}$

P(1000<x<2000) = $e^{-2}-e^{-1}$

P(1000<x<2000) = 0.2325

There is a probability of 23.25% of failure in that interval.

c. Probability of failing between 0 and 1000 hours:

P(0<x<1000) = $e^{\frac{-1000}{1000} }-e^\frac{-0}{1000}$

P(0<x<1000) = $e^{-1}-1$

P(0<x<1000) = 0.6321

There is a probability of 63.21% of failing before 1000 hours.

d. P(x) = $e^{\frac{-b}{1000} }-e^\frac{-a}{1000}$

0.1 = $1-e^\frac{-x}{1000}$

$-e^{\frac{-x}{1000} }=-0.9$

${\frac{-x}{1000} }=ln0.9$

-x = -1000.ln(0.9)

x = 105.4

10% of the components will have failed at 105.4 hours.

5 0

4 years ago

En un videojuego, Marta ha conseguido 36.450 puntos capturando 11 monedas de oro. ¿ cuántos puntos vale cada moneda de oro?

sukhopar [10]

Answer:

3313.64 puntos

Step-by-step explanation:

Podemos interpretar la pregunta anterior Matemáticamente como:

11 monedas de oro = 36.450 puntos

1 moneda de oro = x

Multiplicar cruzada

11 monedas de oro × x = 36.450 puntos × 1 monedas de oro

x = 36.450 puntos × 1 monedas de oro / 11

x = 3313.6363636 puntos

Aproximadamente

1 moneda de oro = 3313.64 puntos

3 0

3 years ago

There are 20 cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all t

blagie [28]

Answer:

The answer is C/85,000

Step-by-step explanation:

6 0

2 years ago

What are the first three terms of the sequence modeled by the recursive function an=1/2an-1 when a1=1?

loris [4]

Answer:

Option A is correct that is $1,\frac{1}{2},\frac{1}{4}$

Step-by-step explanation:

We have been given a formula $a_n=\frac{1}{2}\cdot a_n-1$

$a_1=1$

We will put values of n in $a_n=\frac{1}{2}\cdot a_n-1$

when n=2 we get

$a_2=\frac{1}{2}\cdot a_2-1$

$\Rightarrow a_2=\frac{1}{2}\cdot a_1$

$\Rightarrow a_2=\frac{1}{2}\cdot 1$

$\Rightarrow a_2=\frac{1}{2}$

when n=3

$a_3=\frac{1}{2}\cdot a_2$

$\Rightarrow a_3=\frac{1}{2}\cdot \frac{1}{2}$

$\Rightarrow a_3=\frac{1}{4}$

Therefore, Option A is correct that is $1,\frac{1}{2},\frac{1}{4}$

6 0

3 years ago

Read 2 more answers

Venita is factoring the expression 32ab-8b. She determines the GCF and writes the factored expression as 8b(4a-0). Which best de

Nimfa-mama [501]

Let's go through the steps of factoring that Venita should take.

1.) Find the greatest common factor (GCF). We only have two terms, so that makes it pretty easy.
32 = 1, 2, 4, 8, 16, 32
8 = 1, 2, 4, 8
The greatest common factor of 32 and 8 is 8. We can also factor out a <em>b</em> since that term appears in each part of the original expression. The GCF and variable should go on the outside of the parentheses.
8b( )

2.) Now let's figure out what should go in the middle of the parentheses. To do this, use the original expression and divide each term. This is written in the parentheses.

32ab ÷ 8b = 4a
8b ÷ 8b = 1

This would then result in the factored expression 8b(4a - 1). You can always check this by using the distributive property. Distribute 8b out to both expressions:
8b x 4a = 32ab
8b x 1 = 8b
32ab - 8b is the expression she started with, so your factored expression works!

Now that we went through the steps to solve the factored expression, let's check her answer. The only difference between Venita's and ours is that she has 0 as the second term while we have a 1. It seems that she had subtracted the GCF from the second term instead of dividing.

6 0

3 years ago

Read 2 more answers