If random variable X has a probability density function of f(x)=1/x on the interval[e,e^2], what is the standard deviation of X?

Answer 1

#3.3361599515537#

if the PDF of #x# is #f(x)=1/x# on the interval of #[e,e^2]# then #int_e^(e^2) f(x) dx= 1#

The standard deviation expected is provided by

#E[sigma]=sqrt(int_e^(e^2) (x-mu)^2 f(x)dx )#
#=sqrt(int_e^(e^2) (x^2-2xmu+mu^2)/xdx)#
the integral being #x^2/2 -2mux+mu^2ln(x) #
and #E[sigma] =sqrt( (e^4-e^2)/2 + mu^2 + 2mu - 2mu^2)#
now we need to solve for #mu# and we shall have our final answer
#E[mu]=int_e^(e^2) xf(x)dx #
#=int_e^(e^2) 1/x*x dx #
#=e^2-e = 4.6707742704716#
#E[sigma] = 3.3361599515537#
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Answer 2

To find the standard deviation of a continuous random variable with a probability density function f(x)=1x f(x) = \frac{1}{x} on the interval [e,e2] [e, e^2] , we first need to find the mean, denoted by μ \mu , and then use it to calculate the standard deviation.

The mean μ \mu of a continuous random variable is given by the integral of x x times the probability density function f(x) f(x) over the interval [e,e2] [e, e^2] , divided by the integral of f(x) f(x) over the same interval:

μ=ee2x1xdxee21xdx\mu = \frac{\int_{e}^{e^2} x \cdot \frac{1}{x} dx}{\int_{e}^{e^2} \frac{1}{x} dx}

=ee2dxee21xdx= \frac{\int_{e}^{e^2} dx}{\int_{e}^{e^2} \frac{1}{x} dx}

=e2eln(e2)ln(e)= \frac{e^2 - e}{\ln(e^2) - \ln(e)}

=e2e2ln(e)ln(e)= \frac{e^2 - e}{2\ln(e) - \ln(e)}

=e2e2= \frac{e^2 - e}{2}

Now, the standard deviation σ \sigma can be calculated using the formula:

σ=ee2(xμ)21xdx\sigma = \sqrt{\int_{e}^{e^2} (x - \mu)^2 \cdot \frac{1}{x} dx}

=ee2(xe2e2)21xdx= \sqrt{\int_{e}^{e^2} \left(x - \frac{e^2 - e}{2}\right)^2 \cdot \frac{1}{x} dx}

This integral will need to be evaluated, but first, we need to simplify the expression inside the integral by expanding (xe2e2)2 (x - \frac{e^2 - e}{2})^2 . Once we simplify and integrate, we can then take the square root to find σ \sigma .

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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