Prove that for independent random variables X1, X2, ..., Xn, Var(X1 + X2 + ... + Xn) = Var(X1) + Var(X2) + ... + Var(Xn), where Var(X) := E((X - E(X))^2). Use induction to prove for n = 2. Provide a full solution.
Added by Daniela J.
Step 1
For n=2, we need to prove that Var(X1+X2) = Var(X1) + Var(X2). Show more…
Show all steps
Close
Your feedback will help us improve your experience
Madhur L and 78 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
3. Prove that if X ~ t(n), then X^2 ~ F(1, n) 4. Prove that if X ~ F(m, n), then 1/X ~ F(n, m) 5. Let X1, ... , Xm be iid N(μ1, σ^2) and Y1, ... , Yn be iid N(μ2, σ^2) where Xi and Yj are independent for any i and j. When σ̂^2 = (∑_{i=1}^m (Xi - X̄)^2 + ∑_{j=1}^n (Yi - Ȳ)^2) / (n + m - 2), show (i) that (n + m - 2) * (σ̂^2 / σ^2) ~ χ^2(m + n - 2), (ii) that E(σ̂^2) = σ^2, (iii) and that ((X̄ - Ȳ) - (μ1 - μ2)) / (σ̂ * √(1/m + 1/n)) ~ t(m + n - 2).
Dominador T.
Let X be a random variable and let EX = μ show that (a) E(X - μ)^2 = E(X^2) - μ^2. (b) Var (aX + b) = a^2 Var (X) If X ~ N(μ, σ^2), show (X - μ) / σ ~ N(0,1).
Aman G.
Prove the general case of Theorem 7 . That is, show that if $X_{1}, \quad X_{2}, \ldots, X_{n}$ are pairwise independent random variables on a sample space $S,$ where $n$ is a positive integer, then $V\left(X_{1}+X_{2}+\cdots+X_{n}\right)=$ $V\left(X_{1}\right)+V\left(X_{2}\right)+\cdots+V\left(X_{n}\right) $. [Hint: Generalize the proof given in Theorem 7 for two random variables. Note that a proof using mathematical induction does not work; see Exercise 33.1]
Discrete Probability
Expected Value and Variance
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD