Chapter 6: Q. 22 (page 356)

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) role="math" localid="1649593908596" $P (Y < 0.4)$
(c) role="math" localid="1649593922659" $P (0.1 < Y \leq 0.15 o r 0.77 \leq Y < 0.88$

Short Answer

Expert verified

(a) The probability for $P (Y \leq 0.4)$ is $0.40$

(b) The probability for $P (Y < 0.4$ is $0.40$

Step by step solution

Part (a) Step 1: Given information

To find the probability for $P (Y \leq 0.4)$ .

Part (a) Step 2: Explanation

The variable $Y$ is following the uniform distribution.
The probability for $P (Y \leq 0.4)$ can be determined as:
$P (Y \leq 0.4) = (0.4 - 0) \times 1 = 0.40$

Part (b) Step 1: Given information

To find the probability for $P (Y < 0.4)$

Part (c) Step 2: Explanation

The variable $Y$ is following the uniform distribution.
The probability for $P (Y < 0.4)$ can be determined as:
$P (Y < 0.4) = (0.4 - 0) \times 1 = 0.40$

Part (c) Step 1: Given information

To find the probability for $P (0.1 < Y \leq 0.15 o r 0.77 \leq Y < 0.88)$ .

Part (c) Step 2: Explanation

The probability for $P (0.1 < Y \leq 0.15 o r 0.77 \leq Y < 0.88)$ can be determined as:
$P (0.1 < Y \leq 0.15) = (0.15 - 0.1) \times 1 = 0.05$
$P (0.77 \leq Y \leq 0.88) = (0.88 - 0.77) \times 1 = 0.11$