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
Welcome back — been a while! Let's pick up where you left off. Here's how to solve the problem: Step 1: Set up equations for a and b. The sum of all probabilities in a distribution must equal 1: P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) = 1 a + a + a + b + b = 1 3a + 2b = 1 (Equation 1) The given condition is P(X ≥ 2) = 3P(X < 2). Calculate P(X ≥ 2): P(X ≥ 2) = P(X=2) + P(X=3) + P(X=4) = a + b + b = a + 2b Calculate P(X < 2): P(X < 2) = P(X=0) + P(X=1) = a + a = 2a Substitute these into the given condition: a + 2b = 3(2a) a + 2b = 6a 2b = 5a (Equation 2) Step 2: Solve for a and b. Substitute Equation 2 into Equation 1: 3a + (5a) = 1 8a = 1 a = (1)/(8) Now substitute the value of a back into Equation 2 to find b: 2b = 5 ((1)/(8)) 2b = (5)/(8) b = (5)/(16) a) The values of the probabilities a and b are: a = (1)/(8), b = (5)/(16) Step 3: Calculate the expected value of X, E(X). The formula for the expected value is E(X) = x · P(X=x). E(X) = (0 · a) + (1 · a) + (2 · a) + (3 · b) + (4 · b) E(X) = 0 + a + 2a + 3b + 4b E(X) = 3a + 7b Substitute the values of a = (1)/(8) and b = (5)/(16): E(X) = 3 ((1)/(8)) + 7 ((5)/(16)) E(X) = (3)/(8) + (35)/(16) E(X) = (6)/(16) + (35)/(16) E(X) = (41)/(16) b) The expected value of X is: E(X) = (41)/(16) Step 4: Calculate the standard deviation of X, _X. First, calculate E(X^2): E(X^2) = x^2 · P(X=x) E(X^2) = (0^2 · a) + (1^2 · a) + (2^2 · a) + (3^2 · b) + (4^2 · b) E(X^2) = 0 + a + 4a + 9b + 16b E(X^2) = 5a + 25b Substitute the values of a = (1)/(8) and b = (5)/(16): E(X^2) = 5 ((1)/(8)) + 25 ((5)/(16)) E(X^2) = (5)/(8) + (125)/(16) E(X^2) = (10)/(16) + (125)/(16) E(X^2) = (135)/(16) Next, calculate the variance, Var(X) = E(X^2) - [E(X)]^2: Var(X) = (135)/(16) - ((41)/(16))^2 Var(X) = (135)/(16) - (1681)/(256) Var(X) = (135 · 16)/(256) - (1681)/(256) Var(X) = (2160)/(256) - (1681)/(256) Var(X) = (479)/(256) Finally, calculate the standard deviation, _X = sqrt(Var(X)): _X = sqrt((479)/(256)) _X = sqrt(479)16 c) The standard deviation of X is: _X = sqrt(479)16 Step 5: Calculate the expected value of Y where Y = 5X + 3. Using the property E(cX + d) = cE(X) + d: E(Y) = E(5X + 3) = 5E(X) + 3 Substitute the value of E(X) = (41)/(16): E(Y) = 5 ((41)/(16)) + 3 E(Y) = (205)/(16) + 3 E(Y) = (205)/(16) + (48)/(16) E(Y) = (253)/(16) d) The expected value of Y where Y = 5X + 3 is: E(Y) = (253)/(16) Send me the next one 📸