Probability Distributions Uniform Normal and Exponential Problems

Published on October 3, 2026

Statistics

Step-by-step solutions for uniform distribution, normal distribution probabilities and expected cost, and memoryless property of exponential distribution.

Question

Give me the answers for the above given questions

Answer

Since the image contains multiple problems, I will provide solutions for the distinct mathematical problems presented.

Problem 1 & 3 (Identical)

A random variable

has a uniform distribution over

. The probability density function (pdf) is

for

, and

otherwise.

(i) Compute , , and :

. Since the distribution ends at :

(ii) Find for which :

Problem 2

Time

(i) Proportions:

(a) Less than 35 minutes: .
Using standard normal tables, (or ).
(b) More than 48 minutes: .
(or ).

(ii) Average charge:
Let

be the charge.

Problem 4 & 5 (Identical)

Repair time

is exponentially distributed with parameter

. The pdf is

for

. The survival function is

(i) Probability repair time exceeds 2 hrs:

(ii) Conditional probability :
Due to the memoryless property of the exponential distribution,

.
Here,

and

(iii) Characteristics of Normal Distribution (from Problem 4):

Symmetry: It is bell-shaped and symmetric about the mean .
Measures of Central Tendency: Mean, median, and mode are all equal.
Asymptotic: The tails approach the horizontal axis but never touch it.
Empirical Rule: Approximately of data falls within , within , and within .

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