Solved: Suppose that Y is a continuous random variable

Chapter 4, Problem 191SE

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QUESTION:

Suppose that 𝑌 is a continuous random variable with distribution function given by \(F(y)\) and

probability density function \(f(y)\). We often are interested in conditional probabilities of the form \(P(Y \leq y \mid Y \geq c)\) for a constant 𝑐.

a Show that, for \(y \geq c\),

                                           

                                              \(P(Y \leq y \mid Y \geq c)=\frac{F(y)-F(c)}{1-F(c)}\).

b Show that the function in part (a) has all the properties of a distribution function.

c If the length of life 𝑌 for a battery has a Weibull distribution with \(m=2\) and \(\alpha=3\) (with

measurements in years), find the probability that the battery will last less than four years,

given that it is now two years old.

Equation Transcription:

Text Transcription:

F(y)

f(y)

P(Y</=y|Y>/=c)

y>/=c

P(Y<=y|Y>=c)=F(y)-F(c) over 1-F(c)

m=2

alpha=3

Questions & Answers

QUESTION:

Suppose that 𝑌 is a continuous random variable with distribution function given by \(F(y)\) and

probability density function \(f(y)\). We often are interested in conditional probabilities of the form \(P(Y \leq y \mid Y \geq c)\) for a constant 𝑐.

a Show that, for \(y \geq c\),

                                           

                                              \(P(Y \leq y \mid Y \geq c)=\frac{F(y)-F(c)}{1-F(c)}\).

b Show that the function in part (a) has all the properties of a distribution function.

c If the length of life 𝑌 for a battery has a Weibull distribution with \(m=2\) and \(\alpha=3\) (with

measurements in years), find the probability that the battery will last less than four years,

given that it is now two years old.

Equation Transcription:

Text Transcription:

F(y)

f(y)

P(Y</=y|Y>/=c)

y>/=c

P(Y<=y|Y>=c)=F(y)-F(c) over 1-F(c)

m=2

alpha=3

ANSWER:

Solution 191SE

Step1 of 4:

Let us consider a random variable Y with the distribution function is given by and the probability density function is The conditional probabilities of the form P(Y y|Y c) for a constant c.

Here our goal is:

a). We need to show that, for y ≥ c,

b). We need to show that the function in part (a) has all the properties of a distribution function.

c). We need to find the probability that the battery will last less than four years, given that it is now two years old when the length of life Y for a battery has a Weibull distribution with m = 2 and α = 3.


Step2 of 4:

a).

Let us consider a conditional probability

Now,

Hence the proof.


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