4-4 Density Function of the Continuous Random Variable: practice questions
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 72 questions on this lesson, each with an explanation.
If X is a continuous random variable whose probability density function is f(x), then P(2 < X < 5) equals ……
Athe area of the region beneath the curve of f and above the x-axis from the start of the domain up to x = 2
Bthe area of the region beneath the curve of f and above the x-axis between x = 2 and x = 5
Cf(5) − f(2)
Df(2) + f(5)
Show answer and explanation
Correct answer: (B) the area of the region beneath the curve of f and above the x-axis between x = 2 and x = 5
Explanation: Step 1: The book finds P(a < X < b) as the area of the shaded part beneath the curve of f and above the x-axis between the two values a and b.
Step 2: A sum or a difference of values of the function is not an area.
∴ P(2 < X < 5) is the area beneath the curve of f and above the x-axis between x = 2 and x = 5.
Question 2Easy
If X is a continuous random variable whose probability density function is f(x) = 1/6 where −3 < x < 3, and f(x) = zero otherwise, then P(−1 < X < 2) = ……
A1/2
B2/3
C1/6
D1/3
Show answer and explanation
Correct answer: (A) 1/2
Explanation: Step 1: The region is a rectangle of length 2 − (−1) = 3 and height 1/6.
Step 2: Area = 3 × 1/6 = 1/2
∴ P(−1 < X < 2) = 1/2 (1/6 takes the length as 2 − 1 = 1 by ignoring the sign, and 1/3 takes the length as 2).
Question 3Easy
Which of the following is a discrete random variable?
AThe wage of a government worker chosen at random
BThe height of a player chosen at random from a basketball team
CThe temperature expected during a day
DThe number of absent students in a class on a certain day
Show answer and explanation
Correct answer: (D) The number of absent students in a class on a certain day
Explanation: Step 1: The number of absent students takes countable values 0, 1, 2, …, so it is a discrete random variable.
Step 2: Height, temperature and wage are the book's examples of continuous random variables, whose range is an interval of real numbers.
∴ The number of absent students is the discrete random variable.
Question 4Easy
The point (x, y) lies inside or on the circle x² + y² = 9. If the random variable X expresses the distance of the point from the centre of the circle, then the range of X is ……
A[0, 6]
B[0, 3]
C[−3, 3]
D[0, 9]
Show answer and explanation
Correct answer: (B) [0, 3]
Explanation: Step 1: The radius of the circle is √9 = 3.
Step 2: The distance of any point inside or on the circle from the centre is from 0 to 3.
∴ The range of X = [0, 3] (9 is the square of the radius, 6 is the diameter, and a distance cannot be negative).
Question 5Easy
If X is a continuous random variable, then the relation between P(1 ≤ X ≤ 3) and P(1 < X < 3) is ……
AP(1 ≤ X ≤ 3) = P(1 < X < 3)
BP(1 ≤ X ≤ 3) < P(1 < X < 3)
CP(1 ≤ X ≤ 3) = P(1 < X < 3) + 1
DP(1 ≤ X ≤ 3) > P(1 < X < 3)
Show answer and explanation
Correct answer: (A) P(1 ≤ X ≤ 3) = P(1 < X < 3)
Explanation: Step 1: For a continuous random variable the probability of any single value is zero: P(X = 1) = P(X = 3) = 0.
Step 2: Including or leaving out the end points of the interval does not change the area.
∴ P(1 ≤ X ≤ 3) = P(1 < X < 3).
Question 6Easy
Which of the following is a continuous random variable?
AThe number of loaves of bread a bakery produces in an hour
BThe number of traffic violations recorded on a road in a day
CThe time a student spends waiting for the bus
DThe number of goals a team scores in a match
Show answer and explanation
Correct answer: (C) The time a student spends waiting for the bus
Explanation: Step 1: A continuous random variable has a range that is an interval of real numbers (a non-countable set), such as time, length or temperature.
Step 2: Numbers of goals, loaves and violations take countable values, so they are discrete random variables.
∴ The waiting time is a continuous random variable.
Question 7Medium
Which of the following functions can be a probability density function of a continuous random variable X on the interval 0 < x < 2 (with f(x) = zero otherwise)?
Af(x) = (2x − 1)/2
Bf(x) = x/4
Cf(x) = (x + 1)/2
Df(x) = (2 − x)/2
Show answer and explanation
Correct answer: (D) f(x) = (2 − x)/2
Explanation: Step 1: Both conditions must hold: f(x) ≥ 0 and total area = 1. (2 − x)/2 is positive on the interval, and its area is a triangle: 1/2 × 2 × 1 = 1.
Step 2: (2x − 1)/2 has area 1 but is negative when x < 0.5, x/4 has area 1/2, and (x + 1)/2 has area 2.
∴ f(x) = (2 − x)/2 is a probability density function.
Question 8Hard
A point is chosen at random on a line segment of length 10 cm, dividing it into two parts. The distance of the point from one end of the segment is a continuous random variable X whose probability density function is f(x) = 1/10 where 0 < x < 10, and f(x) = zero otherwise. The probability that the longer part is more than 3 times the shorter part is ……
A1/4
B1/2
C3/4
D1/3
Show answer and explanation
Correct answer: (B) 1/2
Explanation: Step 1: The longer part is more than 3 times the shorter part when the shorter part is less than 10/4 = 2.5 cm, that is, when X < 2.5 or X > 7.5.
Step 2: P = 2.5 × 1/10 + 2.5 × 1/10 = 1/2
∴ The probability is 1/2 (1/4 takes only one end, X < 2.5).
Question 9Hard
X is a continuous random variable whose probability density function is f(x) = (2x + 1)/12 where 0 < x < 3, and f(x) = zero otherwise. The experiment is repeated independently until a value X < 1 occurs for the first time. The expected number of repetitions is ……
A4
B30
C6
D12
Show answer and explanation
Correct answer: (C) 6
Explanation: Step 1: S = P(X < 1) = 1/2 × (f(0) + f(1)) × 1 = 1/2 × (1/12 + 3/12) = 1/6.
Step 2: The number of repetitions is a geometric variable, G(1/6), so its expectation = 1/S = 6.
∴ The expected number of repetitions is 6 (4 comes from taking f(1) = 1/4 as the probability, and 30 is the variance).
Question 10Hard
A bus arrives at a stop every 20 minutes, and a student's waiting time for it (in minutes) is a continuous random variable X whose probability density function is f(x) = 1/20 where 0 < x < 20, and f(x) = zero otherwise. If the student goes to the stop on 3 independent days, the probability that he waits more than 15 minutes on exactly one of these days is ……
A1/4
B9/64
C27/64
D37/64
Show answer and explanation
Correct answer: (C) 27/64
Explanation: Step 1: S = P(X > 15) = 5 × 1/20 = 1/4.
Step 2: The number of such days Y ~ B(3, 1/4): P(Y = 1) = ³C₁ × (1/4) × (3/4)² = 27/64.
∴ The probability is 27/64 (37/64 is the probability of at least one such day, and 9/64 is the probability of exactly two such days).