3.The distribution function of a random variable is given by
F(x)= {0 , for x<-1
{(x+1)/2 , for -1<x<1
{1 , for x≥1.
Find P(3<x<4).
To find P(3 < x < 4), we need to evaluate the distribution function for these values of x and subtract the probability of x < 3 from it.
Given that the random variable x follows the distribution function F(x), we can see that F(-1) = 0 and F(1) = 1.
We know that for -1 < x < 1, the distribution function F(x) = (x + 1) / 2.
Since the given range of x is 3 < x < 4, this lies entirely in the region where x ≥ 1. Therefore, F(x) = 1 for these values of x.
So, to find P(3 < x < 4), we need to calculate F(4) - F(3).
F(4) = 1 (since x ≥ 1)
F(3) = 1 (since x ≥ 1)
Therefore, P(3 < x < 4) = F(4) - F(3) = 1 - 1 = 0.
Hence, the probability of 3 < x < 4 is 0.