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# Density Function of Random Variables

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The destiny function for a random variable X is given in terms of constant c.....

https://brainmass.com/statistics/probability/density-function-random-variables-328295

#### Solution Preview

Since the integral of density function over the domain of X is equal to 1, we can find the value of c by
∫_(-∞)^∞▒〖δ(x)dx=1〗, or
∫_(-∞)^0▒〖ce^x dx+∫_0^∞▒〖ce^(-x) dx〗=1〗
ce^x |_(-∞)^0-ce^(-x) |_0^∞=1
c+c=1
So c=1/2
The distribution function is
If x<0,
∫_(-∞)^x▒〖1/2 e^t dt=1/2 e^t |_(-∞)^x 〗=1/2 e^x
If x≥0
∫_(-∞)^0▒〖1/2 e^t dt+∫_0^x▒〖ce^(-t) dt〗=1/2-1/2 e^(-t) |_0^x 〗=1-1/2 e^(-x)
In general format, the distribution function is
F(x)={█(〖1/2 e〗^x, x<0@1-1/2 e^(-x),x≥0)┤
P(X>1)=1-P(X≤1)=1-F(1)=1-1/2 e^(-1)=1-1/2e
P(-3<X≤4)=F(4)-F(-3)=1-1/2 e^(-4)-1/2 e^(-3)
P(X<-2)=F(-2)=1/2 e^(-2)=1/(2e^2 )
The graph of density function is

The graph of distribution function is
...

#### Solution Summary

The density function of random variables is examined.

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