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I had a hard time to understand Question 59 on page 87 from Ross's book (Introduction to Probability Models)

Let X1,X2,X3,X4 are independent continuous random variables with a common distribution function F and let

p=P(X1 < X2 > X3 < X4)

(a) Argue that the value of p is the same for all continuous distribution functions F.

The Solution says:

(a) Use the fact that F(Xi) is a uniform (0,1) random variable to obtain p=P{F(X1) < F(X2) > F(X3) < F(X4)} =P{U1 < U2 > U3 < U4} where the Ui,i = 1,2,3,4, are independent uniform (0,1) random variables.

My question:

  1. Why "Use the fact that F(Xi) is a uniform (0,1) random variable"? What if F(Xi) is an exponential distribution?
  2. Why "p=P{F(X1) < F(X2) > F(X3) < F(X4)} =P{U1 < U2 > U3 < U4} where the Ui,i = 1,2,3,4, are independent uniform (0,1) random variables"?

Similar Questions related to "Fx(X)",

Density and Distribution of Fx(X)

two function of two random variables

evergreenhomeland
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1 Answers1

2

Look up the following link.

https://en.wikipedia.org/wiki/Probability_integral_transform

Suppose that a random variable X has a continuous distribution for which the cumulative distribution function (CDF) is $F_X$. Then the random variable $Y$ defined as $Y=F_X(X)$ has a uniform distribution.

I hope the above answers 1.

For 2, note that for an increasing function $g, P[X\geq Y] = P[g(X) \geq g(Y)]$. CDF $F$ is an increasing function. Therefore $X_1, X_2, X_3,X_4$ can be replaced with $F(X_1), F(X_2), F(X_3), F(X_4)$ in the equation, which are uniform random variables $U_1, U_2, U_3, U_4$.

Another Grad student
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