The probability distribution for a random variable X is desc

The probability distribution for a random variable X is described by the density function:

a) What value must k have for the density function to be valid?

b) Determine an expression for the cumulative distribution function of X

Solution

a)

The overall integral of the probability distribution must be 1. Thus,

Integral [k e^-x dx]|(0,oo) = -k e^-x |(0, oo) = [-0] - [-k e^-0] = 1

k = 1 [answer]

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b)

Here,

F(x) = Integral [k e^-x dx] = -k e^-x + C

However, note that the limit of F(x) as x-->oo is 1. Thus,

lim [-k e^-x + C] = 1
x-->oo

Thus,

C = 1

Thus,

F(x) = -e^-x + 1 [ANSWER]

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C)

P(x>=5) = 1 - F(5) = 1 - [-e^-5 + 1] = e^-5 = 0.006737947 [answer]