Unit 4 — Random Variables and Probability Distributions · Statistics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 102 questions on this lesson, each with an explanation.
If the expectation of a discrete random variable is μ = 50 and its standard deviation is σ = 4, then the coefficient of variation = ……
Correct answer: (A) 8%
Explanation: Step 1: Coefficient of variation = σ/μ × 100% Step 2: = 4/50 × 100% ∴ the coefficient of variation = 8% (12.5% divides μ by σ, and 0.08% forgets to multiply by 100).
One of the features of the coefficient of variation is that it ……
Correct answer: (D) is not affected by the measuring units, because it is a pure percentage
Explanation: Step 1: Coefficient of variation = σ/μ × 100%, and σ and μ have the same units, so the units cancel. Step 2: It gives the dispersion as a percentage that is not affected by the measuring units of the phenomenon. ∴ it is not affected by the measuring units, because it is a pure percentage.
The variance σ² of a discrete random variable X whose expectation is μ is given by the relation ……
Correct answer: (A) σ² = Σ xᵣ² × f(xᵣ) − μ²
Explanation: Step 1: From the book: σ² = Σ xᵣ² × f(xᵣ) − μ². ∴ σ² = Σ xᵣ² × f(xᵣ) − μ² (the common slips are forgetting to subtract μ² or using xᵣ instead of xᵣ²).
A bag contains 5 cards: two cards each carry the number 2, two cards each carry the number 4, and one card carries the number 9. A card is drawn at random, and X expresses the number on it. Then the standard deviation σ, to the nearest hundredth, = ……
Correct answer: (B) 2.56
Explanation: Step 1: The distribution: xᵣ: 2, 4, 9 and f(xᵣ): 0.4, 0.4, 0.2. μ = 0.8 + 1.6 + 1.8 = 4.2 Step 2: Σ xᵣ² × f(xᵣ) = 1.6 + 6.4 + 16.2 = 24.2, so σ² = 24.2 − 17.64 = 6.56 ∴ σ = √6.56 ≈ 2.56 (6.56 is the variance, and 2.94 treats the values 2, 4, 9 as equally likely).
A fair coin is tossed twice, and X expresses the number of Heads. Then the expectation μ = ……
Correct answer: (B) 1
Explanation: Step 1: The distribution: xᵣ: 0, 1, 2 and f(xᵣ): 1/4, 2/4, 1/4. Step 2: μ = 0 × 1/4 + 1 × 1/2 + 2 × 1/4 = 1/2 + 1/2 ∴ μ = 1 (1/2 is P(X = 1), not the expectation).
If X is a discrete random variable whose probability distribution is xᵣ: −1, 0, 1 and f(xᵣ): 0.3, 0.4, 0.3, then the variance σ² = ……
Correct answer: (A) 0.6
Explanation: Step 1: μ = −0.3 + 0 + 0.3 = 0 and Σ xᵣ² × f(xᵣ) = 0.3 + 0 + 0.3 = 0.6 Step 2: σ² = 0.6 − 0² ∴ σ² = 0.6 (0 mixes up the expectation with the variance, and 0.77 is the standard deviation).
In a competition, a contestant wins 20 pounds with probability 0.1, wins k pounds with probability 0.3, and wins nothing otherwise. If the expectation of what the contestant wins is 5 pounds, then k = ……
Correct answer: (C) 10
Explanation: Step 1: xᵣ: 20, k, 0 and f(xᵣ): 0.1, 0.3, 0.6. Step 2: μ = 20 × 0.1 + 0.3k + 0 = 2 + 0.3k = 5, so 0.3k = 3 ∴ k = 10 (16.67 leaves out the term 20 × 0.1, and 30 divides by 0.1 instead of 0.3).
A box contains n cards numbered from 1 to n. A card is drawn at random, and X expresses the number written on it. If the expectation μ = 5, then the variance σ² = ……
Correct answer: (D) 20/3
Explanation: Step 1: μ = (1 + 2 + … + n)/n = (n + 1)/2 = 5, so n = 9 and each value has probability 1/9 Step 2: Σ xᵣ² × f(xᵣ) = (1 + 4 + … + 81)/9 = 285/9 = 95/3 ∴ σ² = 95/3 − 25 = 20/3 (95/3 forgets to subtract μ², and 33/4 takes n = 10).
The random variable X has the distribution xᵣ: 1, 2, 3 and f(xᵣ): 0.25, 0.5, 0.25, and the random variable Y takes the values 10, 20 with probabilities 0.5, 0.5. Which of the following statements is true?
Correct answer: (A) The standard deviation of Y is greater, but the coefficient of variation of X is greater.
Explanation: Step 1: For X: μ = 2, σ² = 4.5 − 4 = 0.5, σ ≈ 0.71, coefficient of variation ≈ 0.71/2 × 100% ≈ 35.4% Step 2: For Y: μ = 15, σ² = 250 − 225 = 25, σ = 5, coefficient of variation = 5/15 × 100% ≈ 33.3% ∴ σ of Y is greater, but the relative dispersion of X is greater (the two coefficients of variation are not equal, 35.4% against 33.3%; 25% is a trap: it divides the variance, not the standard deviation, by μ).
A family has three children. The probability of having a boy equals the probability of having a girl (twins are not considered). If X expresses the number of boys, then the variance σ² = ……
Correct answer: (C) 0.75
Explanation: Step 1: The distribution: xᵣ: 0, 1, 2, 3 and f(xᵣ): 1/8, 3/8, 3/8, 1/8, so μ = 12/8 = 1.5 Step 2: Σ xᵣ² × f(xᵣ) = (0 + 3 + 12 + 9)/8 = 3 ∴ σ² = 3 − 2.25 = 0.75 (1.5 is the expectation, 3 forgets to subtract μ², and 0.87 is the standard deviation).
The first 10 questions in every subject are free, each with an explanation.