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Magazine Chapter 6: Problem 22
Random numbers Let Y be a number between 0 and 1 produced by a random number generator. Assuming that the random variable Y has a uniform distribution, find the following probabilities: (a) \(P(Y \leq 0.4)\) (b) \(P(Y < 0.4)\) (c) \(P(0.1 < Y \leq 0.15 \text { or } 0.77 \leq Y < 0.88)\)
Short Answer
Expert verified
(a) 0.4
(b) 0.4
(c) 0.16
Step by step solution
01
Understand Uniform Distribution
The uniform distribution for a random variable between 0 and 1 is a simple, equal-probability distribution where each outcome within the interval [0,1] is equally likely. Therefore, the probability density function (pdf) of a uniform distribution U(0,1) is essentially a constant function with a value of 1 over the interval [0,1]. The total area under the curve (representing the entire probability) is always 1.
02
Calculate Probability for Part (a)
Since the distribution is uniform, the probability of a random variable being less than or equal to a certain value is simply that value. So, for part (a), the probability that Y is less than or equal to 0.4 is exactly 0.4. Therefore, \[ P(Y \leq 0.4) = 0.4 \]
03
Calculate Probability for Part (b)
For uniform distributions, the distinction between "less than or equal to" and "less than" does not affect the probability since the probability of hitting any specific exact point is zero due to its continuous nature. Thus, we have:\[ P(Y < 0.4) = 0.4 \]
04
Calculate Probability for Part (c)
For part (c), we need to calculate the probability that Y is within two disjoint intervals: (0.1, 0.15] and [0.77, 0.88).1. For the first interval (0.1, 0.15): The length of the interval is 0.15 - 0.1 = 0.05.2. For the second interval [0.77, 0.88): The length of the interval is 0.88 - 0.77 = 0.11.The total probability is the sum of the probabilities for the two intervals:\[ P(0.1 < Y \leq 0.15 \text{ or } 0.77 \leq Y < 0.88) = 0.05 + 0.11 = 0.16 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Uniform Distribution
The concept of a uniform distribution is one of the simplest in the field of probability. It involves a continuous random variable that has an equal chance of taking any value within a given interval. Imagine spreading a deck of cards evenly across a table鈥攅very point on the table can potentially land a card. For a uniform distribution between 0 and 1, there is no value more likely to occur than any other within this range.
- The interval [0,1] in this case represents the full range of possible outcomes for the variable Y.
- The probability distribution is described by a probability density function (pdf) that is constant over the interval.
- Over any sub-interval, the probability is proportional to the length of that interval.
Probability Calculation
Calculating probability for a uniform distribution is straightforward due to its constant nature. The key idea is that probabilities correspond to the lengths of intervals.
- For example, if you want to calculate the probability of the random variable Y taking a value less than or equal to 0.4, you simply consider the interval length from 0 to 0.4. This interval is 0.4 units long.
- Since the probability density function is 1, the probability for any interval equals the interval's length. Hence, for any value like 0.4, the probability is exactly 0.4.
- This applies the same for comparisons "less than" or "less than or equal to" as individual points have zero probability in a continuous setting.
Continuous Random Variable
A continuous random variable is one that can take an infinite number of possibilities within a given range. Think of it like the wind; it can blow at any speed but only within the limits set by nature. Unlike discrete random variables, which can only take distinct, separate values, continuous variables exist on a spectrum.
- The probability of the variable hitting any single, specific point is actually zero. Instead, we deal with probabilities across ranges or intervals.
- In the case of our variable Y, it could be any number between 0 and 1, including but not limited to every fraction and decimal in between.
- This nature supports why the uniform distribution has such characteristics and necessitates unique ways to calculate probabilities involving continuous random variables.
Probability Density Function
The probability density function (pdf) is what defines a continuous probability distribution. It's a critical concept because it describes how probability is distributed over the values that a random variable can take.
- For a uniform distribution, the pdf is constant across the entire range. In our example of Y uniformly distributed between 0 and 1, the pdf equals 1 over this interval.
- The area under the pdf curve across a range provides the probability of the variable falling within that range.
- For uniform distributions, the total area under the pdf across its full range always sums to 1, reflecting the certainty that the variable will indeed land somewhere within this interval.
