![Given the probability mass function P(x=0) = 1/3, P(x=1) = 1/8, and P(x=2) = 1/2, find E[x] and Var[x].](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1777568703979-e65ffc1509e4ac5d.png&w=3840&q=75)
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hey Ðƴŋâmïč̣, good to see you again. Here's the solution to your probability mass function problem. The probability mass function (PMF) for the random variable x is given as: P(x=0) = (1)/(3) P(x=1) = (1)/(8) P(x=2) = (1)/(2) First, let's check if the probabilities sum to 1: (1)/(3) + (1)/(8) + (1)/(2) = (8)/(24) + (3)/(24) + (12)/(24) = (8+3+12)/(24) = (23)/(24). It seems there might be a slight error in the problem statement as the probabilities do not sum to 1. Assuming the problem intends for these to be the given probabilities, we will proceed with them. If the sum was intended to be 1, one of the probabilities would need to be adjusted. For example, if P(x=2) was (13)/(24) instead of (1)/(2), the sum would be 1. However, we will use the values as written. I) Find E[x] and Var[x] Step 1: Calculate the expected value E[x]. The expected value of a discrete random variable is given by E[x] = x · P(x). E[x] = (0 · P(x=0)) + (1 · P(x=1)) + (2 · P(x=2)) E[x] = (0 · (1)/(3)) + (1 · (1)/(8)) + (2 · (1)/(2)) E[x] = 0 + (1)/(8) + 1 E[x] = (1)/(8) + (8)/(8) E[x] = (9)/(8) Step 2: Calculate E[x^2]. The expected value of x^2 is given by E[x^2] = x^2 · P(x). E[x^2] = (0^2 · P(x=0)) + (1^2 · P(x=1)) + (2^2 · P(x=2)) E[x^2] = (0 · (1)/(3)) + (1 · (1)/(8)) + (4 · (1)/(2)) E[x^2] = 0 + (1)/(8) + 2 E[x^2] = (1)/(8) + (16)/(8) E[x^2] = (17)/(8) Step 3: Calculate the variance Var[x]. The variance is given by Var[x] = E[x^2] - (E[x])^2. Var[x] = (17)/(8) - ((9)/(8))^2 Var[x] = (17)/(8) - (81)/(64) To subtract, find a common denominator, which is 64. Var[x] = (17 · 8)/(8 · 8) - (81)/(64) Var[x] = (136)/(64) - (81)/(64) Var[x] = (136 - 81)/(64) Var[x] = (55)/(64) So, for part I: E[x] = (9)/(8) Var[x] = (55)/(64) II) Find Var[5x - 2] Step 4: Use the property of variance Var[ax + b] = a^2 Var[x]. Here, a=5 and b=-2. Var[5x - 2] = 5^2 · Var[x] Var[5x - 2] = 25 · (55)/(64) Var[5x - 2] = (25 · 55)/(64) Var[5x - 2] = (1375)/(64) So, for part II: Var[5x - 2] = (1375)/(64) III) Find Var[7x + 47] Step 5: Use the property of variance Var[ax + b] = a^2 Var[x]. Here, a=7 and b=47. Var[7x + 47] = 7^2 · Var[x] Var[7x + 47] = 49 · (55)/(64) Var[7x + 47] = (49 · 55)/(64) Var[7x + 47] = (2695)/(64) So, for part III: Var[7x + 47] = (2695)/(64) Send me the next one 📸