Easy Solutions to Random Variable, CDF, and PDF Exercises

Let’s break these exercises down in plain English, like explaining a game to a friend! We’re talking about random variables (things that change by chance, like rolling a die), PDFs (rules showing how likely each outcome is), and CDFs (a running total of those chances). The document you gave didn’t spell out all the rules, so I’ll use simple examples and explain every step, including how to say the math symbols out loud. Math will be in neat code blocks, and key info will be highlighted.

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Exercise 1

(a) Is the rule for chances a real one?

Imagine picking a number between 0 and 4, like guessing how many minutes until a bus comes. The rule, called ( f(x) ) (say “f of x”), tells us the chance of each number. Since it’s not given, let’s assume:

f(x) = 1/4 for 0 ≤ x ≤ 4, and 0 otherwise

This means every number from 0 to 4 has the same chance, like a flat line. The symbol ( \leq ) is “less than or equal to.” For this to be a real rule (a PDF, or “probability density function”):

• It can’t be negative (chances can’t be less than zero!).
• All chances together must equal 1 (100%).

Let’s check:

• Negative? No way! ( 1/4 ) is positive (a quarter), and 0 isn’t negative.

• Adds to 1? Picture a rectangle: height is ( 1/4 ), width is 4 (from 0 to 4). Area is height times width:

  Area = 1/4 × 4 = 4/4 = 1

In math, we use ( \int ) (say “integral,” like adding up tiny pieces) from 0 to 4:

∫₀⁴ (1/4) dx = (1/4) × (x)

₀⁴ = (1/4) × (4 - 0) = (1/4) × 4 = 1

Here, ( dx ) (say “dee x”) is a tiny bit of ( x ) (say “x”), and (

₀⁴ ) means “from 0 to 4.”

Key Info: The total chance is 1, and it’s never negative, so it’s a real rule!

Answer: Yes, ( f(x) ) is a valid PDF.

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(b) What’s the chance of picking between 0 and 2?

Using our rule ( f(x) = 1/4 ) from 0 to 4, what’s the chance ( X ) (say “big X”) is between 0 and 2? We’ll do it two ways:

• Picture it (geometrically): The whole chance from 0 to 4 is 1 (the rectangle’s area). From 0 to 2 is half the width (2 out of 4), with height ( 1/4 ):

  Chance = width × height = 2 × (1/4) = 2/4 = 1/2 = 0.5

That’s 50%!

• Math way (integration): We add up the chances from 0 to 2:

  P(0 ≤ X ≤ 2) = ∫₀² (1/4) dx = (1/4) × (x)

  ₀² = (1/4) × (2 - 0) = (1/4) × 2 = 2/4 = 1/2 = 0.5

( P ) (say “P”) means “probability,” and the rest is like before—same answer!

Key Info: Both ways show a 50% chance, so we’re spot on!

Answer: ( P(0 \leq X \leq 2) = 0.5 ).

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Exercise 4

(a) Can this rule work for chances?

No rule’s given, so let’s try:

f(x) = (3/8) x² for 0 ≤ x ≤ 2, and 0 otherwise

This says chances grow as ( x ) gets bigger—like waiting for a bus that’s more likely to come later. ( x² ) (say “x squared”) means ( x ) times itself. Check it:

• Negative? Nope! ( x² ) is 0 or more (e.g., ( 1^2 = 1 )), and ( 3/8 ) (three-eighths) is positive.

• Adds to 1? We need the area under this curve (it’s a hill, not a flat line):

  ∫₀² (3/8) x² dx = (3/8) × (x³ / 3)

  ₀²

( x³ ) (say “x cubed”) is ( x \times x \times x ). Solve it:

= (3/8) × [(2³ / 3) - (0³ / 3)] = (3/8) × (8/3 - 0) = (3/8) × (8/3) = (3 × 8) / (8 × 3) = 24/24 = 1

Key Info: It’s positive and totals 1—perfect!

Answer: Yes, ( f(x) ) is a valid PDF.

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Exercise 5

What’s the running total of chances for ( X ) between 0 and 2?

We want the CDF (say “see-dee-eff”), which tracks the total chance up to ( x ). No rule’s given, so assume:

f(x) = 1/2 for 0 < x < 2, and 0 otherwise

A fair chance from 0 to 2. The CDF is ( F(x) ) (say “F of x”) = chance ( X \leq x ):

• Before 0 (( x \leq 0 )): No chance yet:

  F(x) = 0

• Between 0 and 2 (( 0 < x < 2 )): Chance grows evenly:

  F(x) = ∫₀ˣ (1/2) dt = (1/2) × (t)

  ₀ˣ = (1/2) × (x - 0) = (1/2) × x = x/2

( t ) (say “tee”) is like ( x ), just a stand-in. E.g., at ( x = 1 ), ( F(x) = 1/2 ).

• At 2 or more (( x \geq 2 )): All the chance:

  F(x) = ∫₀² (1/2) dt = (1/2) × (t)

  ₀² = (1/2) × (2 - 0) = (1/2) × 2 = 1

( \geq ) is “greater than or equal to.”

Key Info: The tally starts at 0, climbs to 1, and stays there—makes sense!

Answer:

F(x) = 0 if x ≤ 0 x/2 if 0 < x < 2 1 if x ≥ 2