Mean and varince of Log(X) Where X~U[1,0]

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In summary, the problem asks to find the mean and variance of log(X) where X~U[1,0] and X is a continuous random variable. The equations used are \mathbb{E}(X) = \int_{-\infity}^{\infity}{x f_X(x)} dx, \mathbb{E}(X^2) = \int_{-\infity}^{\infity}{x^2 f_X(x)} dx, and Var(X) = \mathbb{E}(X^2) - (\mathbb{E}(X))^2. The solution involves finding the probability distribution g(y) of Y, where Y = log(X), and using the formula for mean and variance
  • #1
rosh300
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Homework Statement



find the mean and varince of Log(X) Where X~U[1,0] (X is continuous Random variable)

Homework Equations


[tex] \mathbb{E}(X) = \int_{-\infity}^{\infity}{x f_X(x)} dx [/tex]

[tex] \mathbb{E}(X^2) = \int_{-\infity}^{\infity}{x^2 f_X(x)} dx [/tex]

[tex] Var(X) = \mathbb{E}(X^2) - (\mathbb{E}(X))^2 [/tex]

The Attempt at a Solution


[tex] \mathbb{E}[log(x)] = \int_0^1{xlog(x)} = \frac{-1}{4} [/tex]
[tex] \mathbb{E}[log(x)^2] = \int_0^1{x^2log(x)} = \frac{-1}{9} [/tex]
[tex] Var(X) = \mathbb{E}[log(x)^2] - \mathbb{E}[log(x)]^2 = \frac{-25}{144} [/tex]

but i know variance can't be negative
 
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  • #2
you're right, variance can't be negative

not too sure what you've done, but i think you've assumed [itex] f_{log(X)}(log(X)=x) = log(x)[/itex] which doesn't make any sense... in fact over the given domain log(X) is negative, which doesn't make sense for a pdf, and isn't normalised

So first, define the random variable Y, related to X by
Y = log(X)

you then need to find the probabilty distrubution g(y).dy. It can be related to the original f(x), by

g(y).dy = f(x).dx
where the dy & dx are related

once you have that
[tex] E[log(X)] = E[Y] = \int{g(y)}dy [/tex]

and so on, note the lmits of the pdf of y, g(y) wll be realte to the orginal x values, but not necessarily the same
 
Last edited:
  • #3
i think i get it now
let:[tex] f(x) = 1 \mbox{(the pdf for U[0,1]), }g(y) = e^y = x \Rightarrow \frac{dx}{dy} = e^y, f(g(y)) = 1 [/tex]

this gives you [tex]f_x(x) = \int_0^1{1 dx} = \int_{log(0)}^{log(1)}{e^y dy} = \int^0_{-\infty}e^y dy[/tex]
get the pdf fo y from that and use the def/formula for mean and varaince.

it seems so simple thanks
 

What is the meaning of "Mean and variance of Log(X) where X~U[1,0]"?

The phrase "Mean and variance of Log(X) where X~U[1,0]" refers to the calculation of the mean and variance of the natural logarithm of a random variable X that follows a uniform distribution between 0 and 1.

How is the mean of Log(X) calculated?

The mean of Log(X) can be calculated using the formula E[Log(X)] = ∫log(x)f(x)dx, where f(x) is the probability density function of X. For a uniform distribution, the mean of Log(X) is equal to -1.

What is the significance of calculating the variance of Log(X)?

The variance of Log(X) helps to measure the spread of the natural logarithm of a random variable X. It is a useful tool in understanding the variability of data and can be used to make predictions and draw conclusions about the underlying distribution of X.

What is the relationship between the mean and variance of Log(X)?

The mean and variance of Log(X) are related to each other through the formula Var[Log(X)] = E[(Log(X) - E[Log(X)])^2]. This formula shows that the variance is a measure of how much the values of Log(X) deviate from its mean. A higher variance indicates a wider spread of data and vice versa.

How can the mean and variance of Log(X) be used in practical applications?

The mean and variance of Log(X) can be used in various fields such as finance, economics, and statistics. For example, in finance, the mean and variance of Log(X) can be used to calculate the expected return and risk of an investment. In economics, they can be used to analyze the distribution of income or wealth. In statistics, they can be used to make inferences about the population mean and variance based on a sample of data.

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