# Conditional expectation of Exp(theta)

• ghostyc
In summary, to show that \operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta, you can use the memorylessness property and rewrite it in a form suitable for conditional probabilities. By expressing the conditional expectation as a ratio of integrals and evaluating it directly, you can then show the desired result.
ghostyc
Given X follows an exponential distribution $$\theta$$

how could i show something like

$$\operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta$$

?

i have get the idea of using Memorylessness property here,
but how can i combine the probabilty with the expectation?

thanks.

casper

ghostyc said:
i have get the idea of using Memorylessness property here,
but how can i combine the probabilty with the expectation?

If you write down mathematically the Memorynessless property then it might become obvious. Otherwise it's fine to express the conditional expectation as a ratio of integrals and evaluate it directly.

bpet said:
If you write down mathematically the Memorynessless property then it might become obvious. Otherwise it's fine to express the conditional expectation as a ratio of integrals and evaluate it directly.

$$\Pr(T > s + t\; |\; T > s) = \Pr(T > t) \;\; \hbox{for all}\ s, t \ge 0.$$

i just can't convert from expectation to the probability...

damn

That's what I did so far.

But I just can't use the memoryless property to do it ...

http://img138.imageshack.us/img138/5945/tempz.jpg

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ghostyc said:
That's what I did so far.

But I just can't use the memoryless property to do it ...

Hint: rewrite the memoryless property into a form suitable to use on line 1 of your proof. Conditional probabilities -> conditional cdf -> conditional pdf.

## What is the definition of conditional expectation of Exp(theta)?

The conditional expectation of Exp(theta) is the expected value of the exponential distribution with a given value of the parameter theta, given that a certain event has occurred.

## How is the conditional expectation of Exp(theta) calculated?

The conditional expectation of Exp(theta) is calculated by taking the integral of the exponential distribution with respect to theta, multiplied by the conditional probability of the event occurring.

## Why is the conditional expectation of Exp(theta) important in statistics?

The conditional expectation of Exp(theta) is important because it allows us to make predictions about the behavior of the exponential distribution when certain events occur. It also helps us understand the relationship between the parameter theta and the observed data.

## Can the conditional expectation of Exp(theta) be negative?

No, the conditional expectation of Exp(theta) cannot be negative. The exponential distribution is always positive, so the expected value will also be positive.

## How does the conditional expectation of Exp(theta) change with different values of theta?

The conditional expectation of Exp(theta) will change as theta changes. As theta increases, the expected value will also increase, and vice versa.

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