# Question: Show that the jointly continuous discrete random variables X1

Show that the jointly continuous (discrete) random variables X1, . . . ,Xn are independent if and only if their joint probability density (mass) function f (x1, . . . , xn) can be written as

for nonnegative functions gi(x), i = 1, . . . , n.

for nonnegative functions gi(x), i = 1, . . . , n.

## Answer to relevant Questions

In Example 5c we computed the conditional density of a success probability for a sequence of trials when the first n + m trials resulted in n successes. Would the conditional density change if we specified which n of these ...Suppose X and Y are both integer-valued random variables. Let p(i|j) = P(X = i|Y = j) and q(j|i) = P(Y = j|X = i) Show that If X is exponential with rate λ, find P{[X] = n,X − [X] ≤ x}, where [x] is defined as the largest integer less than or equal to x. Can you conclude that [X] and X − [X] are independent? If X and Y are independent standard normal random variables, determine the joint density function of U = X V= X/Y Then use your result to show that X/Y has a Cauchy distribution. The positive random variable X is said to be a lognormal random variable with parameters μ and σ2 if log(X) is a normal random variable with mean μ and variance σ2. Use the normal moment generating function to find the ...Post your question