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Assignment 6: IC252 - IIT Mandi

This document contains instructions for Assignment 6 which is due on April 15, 2022. The assignment contains 6 problems involving probability density functions and random variables. Problem 1 involves finding the constant and probability for a given PDF. Problem 2 checks if a given PDF is valid and finds an underweight probability. Problem 3 identifies whether random variables for height, grade, and plate thickness are discrete or continuous. Problem 4 involves plotting, checking total area, and finding probabilities and statistics for a given PDF. Problem 5 verifies properties for normal distributions. Problem 6 verifies properties for a normal distribution with mean μ and variance σ2.

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319 views2 pages

Assignment 6: IC252 - IIT Mandi

This document contains instructions for Assignment 6 which is due on April 15, 2022. The assignment contains 6 problems involving probability density functions and random variables. Problem 1 involves finding the constant and probability for a given PDF. Problem 2 checks if a given PDF is valid and finds an underweight probability. Problem 3 identifies whether random variables for height, grade, and plate thickness are discrete or continuous. Problem 4 involves plotting, checking total area, and finding probabilities and statistics for a given PDF. Problem 5 verifies properties for normal distributions. Problem 6 verifies properties for a normal distribution with mean μ and variance σ2.

Uploaded by

Suhana Singh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Assignment 6

IC252 - IIT Mandi


Submission Deadline: 15 April, 2022

1. Suppose that X is a continuous r.v. whose PDF is given by


(
c(4x − 2x2 ), 0 ≤ x ≤ 2;
fX (x) =
0, elsewhere.

(a) What is the value of the constant c (for fX to be a valid PDF)? [2]
(b) Find P (X > 1). [1]
2. Milk containers have label printed “2 liters”. But, the PDF of the amount of milk deposited
in a milk container by a dairy factory is
(
40.976 − 16x − 30e−x , 1.95 ≤ x ≤ 2.20;
fX (x) =
0, elsewhere.

(a) Is fX a valid PDF? [2]


(b) What is the probability that a container produced by the dairy factory is underweight?[2]
3. Consider a random variable measuring the following quantities. In each case state with rea-
sons whether you think it more appropriate to define the random variable as discrete or as
continuous.
(a) A person’s height [1]
(b) A student’s course grade [1]
(c) The thickness of a metal plate [1]
4. A random variable X takes values between 4 and 6 with a probability density function
(
1
, 4 ≤ x ≤ 6;
fX (x) = x loge (1.5)
0 elsewhere.

(a) Make a plot of the PDF (you may use some programming tools to plot functions). [1]
(b) Check that the total area under the probability density function is equal to 1. [2]
(c) What is P (4.5 ≤ X ≤ 5.5)? [2]
(d) Find the CDF and plot it (you may use some programming tools to plot functions). [2]
(e) What is the expected value of this random variable? [1.5]

1
(f) What is the median of this random variable? [1.5]
(g) What is the variance of this random variable? [2]
(h) What is the standard deviation of this random variable? [1]
5. (a) For X ∼ N (µ, σ 2 ), verify that, its PDF is symmetric around the mean, i.e., [1]

fX (µ − x) = fX (µ + x).

(b) For X ∼ N (0, 1), verify that [1.5]

ΦX (−x) = 1 − ΦX (x).

6. Optional (advanced): For X ∼ N (µ, σ 2 ), i.e., X with PDF


1 2 2
fX (x) = √ e−(x−µ) /2σ
σ 2π
verify that
(a) the mean is µ and [2.5]
2
(b) the variance is σ . [2.5]

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