Suppose a random variable x has the following probability density function: f(x)=1/x, 1≤x≤c, or f(x)=0 otherwise a) what must the value of c be so that f(x) is a probability density function? b) find p(x< 2)c) find e(x) and var(x)

a)  

b)

c)

Step-by-step explanation:

a) what must the value of C be so that f(x) is a probability density function?

In order to be a probability function we need this condition:

And solving the left part of the integral we have:

, so then

b) find P(X<2)

We can find this probability on this way using the density function:

c) find E(X) and Var(X)

We can find the expected value on this way:

In order to find the Var(X) we need to find the second moment given by:

And now we can use the following definition: