Question 1169914: Suppose the random variable X has a binomial(n,U)distribution where U is uniformly distributed random variable on [0,1].
Show that X is a discrete uniformly distributed random variable.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step by step:
**1. Understand the Distributions:**
* **Binomial Distribution:** X | U = u ~ Binomial(n, u). This means that given a specific value of U = u, X follows a binomial distribution with n trials and probability of success u.
* **Uniform Distribution:** U ~ Uniform(0, 1). This means that U is a continuous random variable with a constant probability density function over the interval [0, 1].
**2. Find the Probability Mass Function (PMF) of X:**
We need to find P(X = k) for k = 0, 1, 2, ..., n.
P(X = k) = ∫ P(X = k | U = u) * f_U(u) du
where:
* P(X = k | U = u) = (n choose k) * u^k * (1 - u)^(n - k) (binomial PMF)
* f_U(u) = 1 (uniform PDF on [0, 1])
So,
P(X = k) = ∫[0, 1] (n choose k) * u^k * (1 - u)^(n - k) * 1 du
P(X = k) = (n choose k) * ∫[0, 1] u^k * (1 - u)^(n - k) du
**3. Recognize the Beta Function:**
The integral ∫[0, 1] u^k * (1 - u)^(n - k) du is related to the Beta function.
The Beta function is defined as:
B(x, y) = ∫[0, 1] t^(x - 1) * (1 - t)^(y - 1) dt
In our case, x = k + 1 and y = (n - k) + 1 = n - k + 1.
So,
∫[0, 1] u^k * (1 - u)^(n - k) du = B(k + 1, n - k + 1)
**4. Relate the Beta Function to Factorials:**
The Beta function has a relation to factorials:
B(x, y) = Γ(x) * Γ(y) / Γ(x + y)
where Γ(z) is the Gamma function. For integer values, Γ(z) = (z - 1)!.
Therefore,
B(k + 1, n - k + 1) = k! * (n - k)! / (n + 1)!
**5. Substitute Back into P(X = k):**
P(X = k) = (n choose k) * [k! * (n - k)! / (n + 1)!]
P(X = k) = [n! / (k! * (n - k)!)] * [k! * (n - k)! / (n + 1)!]
P(X = k) = n! / (n + 1)!
P(X = k) = 1 / (n + 1)
**6. Conclusion:**
Since P(X = k) = 1 / (n + 1) for k = 0, 1, 2, ..., n, X is a discrete uniformly distributed random variable over the set {0, 1, 2, ..., n}.
**Therefore, X is a discrete uniformly distributed random variable.**
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