Statistics for Data Science I — Week 12: Continuous Random Variables & PDFs

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TL;DR

$$P(X=x)$$ $$\boxed{ P(X=x)=0 }$$ $$\boxed{ P(a\leq X\leq b) = \int_a^b f(x)\,dx }$$ $$f(x)$$

Uniform

$$X\sim U(a,b)$$ $$\boxed{ f(x)=\frac1{b-a} }$$ $$a\leq x\leq b.$$

Exponential

$$X\sim\operatorname{Exp}(\lambda)$$ $$\boxed{ f(x)=\lambda e^{-\lambda x}, \qquad x\geq0 }$$ $$\boxed{ P(X>x)=e^{-\lambda x} }$$

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1. Continuous random variable recap

• height
• waiting time
• lifetime
• distance
• temperature
• time between arrivals

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2. Example

$$X=\text{waiting time in minutes}.$$ $$2,\ 2.1,\ 2.01,\ 2.001,\dots$$

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3. Discrete vs continuous

Discrete

$$P(X=x).$$ $$P(X=3)=0.2.$$

Continuous

$$P(a<X<b).$$ $$\boxed{ P(X=x)=0. }$$

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4. Why is point probability zero?

$$\text{area} = \text{height}\times0 = 0.$$ $$P(X=x)=0.$$

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5. Zero probability does not mean impossible

$$X$$ $$P(X=2)=0,$$

an individual point has zero probability mass.

the value is logically impossible.

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6. Probability density function

$$\boxed{ f_X(x) }$$ $$f(x).$$

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7. PDF is not itself probability

$$f(x)$$ $$P(X=x).$$ $$P(X=x)=0$$ $$f(x)>0.$$

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8. Probability as area

$$a<b,$$ $$\boxed{ P(a\leq X\leq b) = \int_a^b f(x)\,dx }$$

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9. PDF conditions

$$\boxed{ f(x)\geq0 }$$ $$\boxed{ \int_{-\infty}^{\infty}f(x)\,dx=1 }$$

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10. PMF vs PDF

PMF

$$p(x)=P(X=x).$$ $$\sum_xp(x)=1.$$

PDF

$$f(x)\neq P(X=x).$$ $$\int_{-\infty}^{\infty}f(x)\,dx=1.$$

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11. Discrete sum vs continuous integral

$$\boxed{ \text{Discrete: } P(X\in A)=\sum_{x\in A}p(x) }$$ $$\boxed{ \text{Continuous: } P(X\in A)=\int_Af(x)\,dx }$$

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12. Endpoints do not matter

$$P(X=a)=0$$ $$P(X=b)=0.$$ $$\boxed{ P(a<X<b) = P(a\leq X<b) = P(a<X\leq b) = P(a\leq X\leq b) }$$

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13. Example

$$P(2<X<5)=0.4,$$ $$P(2\leq X\leq5)=0.4$$

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14. CDF recap

$$\boxed{ F(x)=P(X\leq x) }$$ $$\boxed{ F(x) = \int_{-\infty}^{x}f(t)\,dt }$$

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15. CDF from PDF

$$F(x) = \text{area under }f\text{ to the left of }x.$$

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16. Probability from CDF

$$a<b,$$ $$\boxed{ P(a<X\leq b) = F(b)-F(a) }$$

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17. PDF from CDF

$$\boxed{ f(x)=F'(x) }$$ $$\boxed{ \text{CDF = integral of PDF} }$$ $$\boxed{ \text{PDF = derivative of CDF} }$$

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18. Example PDF

$$f(x)= \begin{cases} 2x,&0\leq x\leq1\\ 0,&\text{otherwise} \end{cases}$$ $$\int_0^1 2x\,dx = [x^2]_0^1 = 1.$$

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19. Probability from that PDF

$$P(X\leq0.5).$$ $$\int_0^{0.5}2x\,dx.$$ $$[x^2]_0^{0.5}$$ $$=(0.5)^2$$ $$=\boxed{0.25}.$$

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20. Another probability

$$P(0.25<X<0.75).$$ $$\int_{0.25}^{0.75}2x\,dx.$$ $$[x^2]_{0.25}^{0.75}$$ $$= 0.75^2-0.25^2$$ $$= 0.5625-0.0625$$ $$= \boxed{0.5}.$$

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21. Finding an unknown constant

$$f(x)= \begin{cases} kx,&0\leq x\leq2\\ 0,&\text{otherwise} \end{cases}$$ $$\int_0^2kx\,dx=1.$$

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22. Solve for $k$

$$k\left[\frac{x^2}{2}\right]_0^2=1.$$ $$k\cdot2=1.$$ $$\boxed{ k=\frac12 }$$

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23. PDF height can exceed 1

A PDF value itself can be greater than 1.

$$0$$ $$1.$$

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24. Example

$$f(x)=2$$ $$0\leq x\leq0.5.$$ $$2(0.5)=1.$$ $$f(x)=2>1.$$

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25. Uniform distribution

$$\boxed{ X\sim U(a,b) }$$ $$a<b.$$

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26. Uniform PDF

$$\boxed{ f(x)= \begin{cases} \dfrac1{b-a},&a\leq x\leq b\\ 0,&\text{otherwise} \end{cases} }$$

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27. Why the height is $\frac1{b-a}$

$$b-a.$$ $$c.$$ $$1.$$ $$c(b-a)=1.$$ $$\boxed{ c=\frac1{b-a} }$$

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28. Uniform probability

$$a\leq c<d\leq b,$$ $$P(c\leq X\leq d) = \frac{d-c}{b-a}.$$ $$\boxed{ P(c\leq X\leq d) = \frac{\text{interval length of interest}} {\text{total interval length}} }$$

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29. Example

$$X\sim U(0,10).$$ $$P(2\leq X\leq5).$$ $$5-2=3.$$ $$10-0=10.$$ $$\boxed{ P(2\leq X\leq5)=\frac3{10}=0.3. }$$

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30. Uniform CDF

$$X\sim U(a,b),$$ $$F(x)= \begin{cases} 0,&x<a\\ \dfrac{x-a}{b-a},&a\leq x\leq b\\ 1,&x>b \end{cases}$$

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31. Uniform expectation

$$\boxed{ E[X] = \frac{a+b}{2} }$$

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32. Uniform variance

$$X\sim U(a,b),$$ $$\boxed{ \operatorname{Var}(X) = \frac{(b-a)^2}{12} }$$

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33. Uniform standard deviation

$$\boxed{ SD(X) = \frac{b-a}{\sqrt{12}} }$$

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34. Uniform intuition

equal-length intervals inside the support have equal probability.

$$X\sim U(0,10),$$ $$[1,2]$$ $$[8,9].$$ $$1.$$

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35. But individual points are still probability zero

$$P(X=5)=0.$$

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36. Exponential distribution

$$\boxed{ X\sim\operatorname{Exp}(\lambda) }$$ $$\lambda>0.$$

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37. Exponential PDF

$$\boxed{ f(x)= \begin{cases} \lambda e^{-\lambda x},&x\geq0\\ 0,&x<0 \end{cases} }$$ $$[0,\infty).$$

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38. Why only nonnegative values?

$$P(X<0)=0.$$

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39. Check exponential PDF

$$\int_0^\infty \lambda e^{-\lambda x}\,dx.$$ $$-e^{-\lambda x}.$$ $$[-e^{-\lambda x}]_0^\infty = 0-(-1) = 1.$$

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40. Exponential CDF

$$x\geq0,$$ $$F(x) = \int_0^x \lambda e^{-\lambda t}\,dt.$$ $$\boxed{ F(x)=1-e^{-\lambda x} }$$ $$x\geq0.$$

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41. Exponential survival function

$$P(X>x) = 1-F(x).$$ $$\boxed{ P(X>x) = e^{-\lambda x} }$$

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42. Example

$$X\sim\operatorname{Exp}(0.5).$$ $$P(X>4) = e^{-0.5(4)}$$ $$= \boxed{e^{-2}}.$$

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43. Probability between two times

$$0\leq a<b,$$ $$P(a<X<b) = F(b)-F(a).$$ $$= (1-e^{-\lambda b}) - (1-e^{-\lambda a}).$$ $$\boxed{ P(a<X<b) = e^{-\lambda a} - e^{-\lambda b} }$$

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44. Exponential expectation

$$X\sim\operatorname{Exp}(\lambda),$$ $$\boxed{ E[X]=\frac1\lambda }$$ $$\lambda$$

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45. Exponential variance

$$\boxed{ \operatorname{Var}(X) = \frac1{\lambda^2} }$$ $$\boxed{ SD(X)=\frac1\lambda }$$

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46. Rate vs waiting time

$$\lambda=5$$ $$\frac15\text{ hour}.$$ $$12\text{ minutes}.$$ $$\boxed{ \text{higher event rate} \rightarrow \text{shorter mean waiting time} }$$

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47. Poisson ↔ exponential connection

number of events in an interval.

waiting time until the next event.

text
Poisson:
How many arrivals in 10 minutes?

Exponential:
How long until the next arrival?

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48. Example connection

$$\lambda=3$$

Number of calls in one hour

$$N\sim\operatorname{Pois}(3).$$

Waiting time until next call

$$T\sim\operatorname{Exp}(3)$$ $$\frac13\text{ hour} = 20\text{ minutes}.$$

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49. Memoryless property

$$\boxed{ P(X>s+t\mid X>s) = P(X>t) }$$

If you have already waited $s$ units, the remaining waiting-time distribution behaves as though the clock restarted.

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50. Example intuition

$$X$$ $$10$$

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51. Algebraic verification

$$P(X>x)=e^{-\lambda x},$$ $$P(X>s+t\mid X>s) = \frac{P(X>s+t)}{P(X>s)}.$$ $$= \frac{ e^{-\lambda(s+t)} }{ e^{-\lambda s} }$$ $$= e^{-\lambda t}.$$ $$e^{-\lambda t} = P(X>t).$$

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52. Uniform vs exponential

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53. Discrete vs continuous distribution summary

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54. PMF / PDF / CDF comparison

PMF

$$p(x)=P(X=x)$$

PDF

$$f(x)$$ $$\int f.$$

CDF

$$F(x)=P(X\leq x)$$

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$$P(X=2)=f(2).$$ $$\boxed{ P(X=2)=0. }$$

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$$\leq1.$$ $$\int f(x)\,dx=1.$$

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$$X\sim U(2,8).$$ $$f(x)=0$$ $$[2,8].$$

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58. Example

$$X\sim U(2,8),$$ $$P(0<X<4).$$ $$[2,4]$$ $$P= \frac{4-2}{8-2} = \boxed{\frac13}.$$

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$$P(X<5)=P(X\leq5).$$

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$$E[X]=\lambda.$$ $$\boxed{ E[X]=\frac1\lambda. }$$

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$$F(x)=1-e^{-\lambda x}.$$ $$\boxed{ P(X>x)=e^{-\lambda x}. }$$

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$$4\text{ events/hour},$$ $$e^{-\lambda x}.$$ $$\lambda x$$

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63. Unit conversion example

$$6\text{ per hour}.$$ $$\frac6{60} = 0.1$$ $$P(X>10) = e^{-0.1(10)} = \boxed{e^{-1}}.$$

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64. Full PDF-normalization example

$$f(x)= \begin{cases} k(1+x),&0\leq x\leq1\\ 0,&\text{otherwise} \end{cases}$$ $$\int_0^1k(1+x)\,dx=1.$$

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65. Solve

$$k\left[ x+\frac{x^2}{2} \right]_0^1 = 1.$$ $$k\left(1+\frac12\right)=1.$$ $$\frac32k=1.$$ $$\boxed{ k=\frac23. }$$

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66. Then compute probability

$$P\left(X\leq\frac12\right).$$ $$= \int_0^{1/2} \frac23(1+x)\,dx.$$ $$= \frac23 \left[ x+\frac{x^2}{2} \right]_0^{1/2}.$$ $$\frac12+\frac18 = \frac58.$$ $$\frac23\cdot\frac58 = \boxed{\frac5{12}}.$$

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67. Continuous probability workflow

text
1. Identify support
2. Check/solve normalization
3. Translate event to an interval
4. Integrate PDF over that interval
5. Use complement/CDF if easier
6. Check answer lies in [0,1]

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68. Uniform workflow

$$X\sim U(a,b),$$ $$\boxed{ P(c<X<d) = \frac{d-c}{b-a} }$$

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69. Exponential workflow

$$X\sim\operatorname{Exp}(\lambda),$$ $$F(x)=1-e^{-\lambda x}$$ $$P(X>x)=e^{-\lambda x}.$$

Less than

$$\boxed{ P(X<x)=1-e^{-\lambda x} }$$

Greater than

$$\boxed{ P(X>x)=e^{-\lambda x} }$$

Between

$$\boxed{ P(a<X<b) = e^{-\lambda a}-e^{-\lambda b} }$$

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70. Full Stats 1 probability progression

text
Weeks 5–6
Counting
   ↓
Week 7
Probability
   ↓
Week 8
Conditional probability + Bayes
   ↓
Week 9
Random variables + PMF/CDF
   ↓
Week 10
Expectation + variance
   ↓
Week 11
Bernoulli / Binomial / Poisson
   ↓
Week 12
Continuous RVs / PDF
├── Uniform
└── Exponential

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71. Discrete–continuous bridge

$$\boxed{ P(X=k) }$$ $$\boxed{ P(X=x)=0 }$$ $$\boxed{ P(a<X<b) = \int_a^bf(x)\,dx. }$$

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72. Formula sheet — continuous RVs

PDF conditions

$$\boxed{ f(x)\geq0 }$$ $$\boxed{ \int_{-\infty}^{\infty}f(x)\,dx=1 }$$

Interval probability

$$\boxed{ P(a<X<b) = \int_a^bf(x)\,dx }$$

CDF

$$\boxed{ F(x) = \int_{-\infty}^{x}f(t)\,dt }$$

Density from CDF

$$\boxed{ f(x)=F'(x) }$$

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73. Formula sheet — uniform

$$X\sim U(a,b),$$ $$\boxed{ f(x)=\frac1{b-a} }$$ $$\boxed{ P(c<X<d) = \frac{d-c}{b-a} }$$ $$\boxed{ E[X]=\frac{a+b}{2} }$$ $$\boxed{ \operatorname{Var}(X) = \frac{(b-a)^2}{12} }$$

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74. Formula sheet — exponential

$$X\sim\operatorname{Exp}(\lambda),$$ $$\boxed{ f(x)=\lambda e^{-\lambda x}, \quad x\geq0 }$$ $$\boxed{ F(x)=1-e^{-\lambda x} }$$ $$\boxed{ P(X>x)=e^{-\lambda x} }$$ $$\boxed{ E[X]=\frac1\lambda }$$ $$\boxed{ \operatorname{Var}(X)=\frac1{\lambda^2} }$$

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75. What you should actually master

1. Distinguish discrete and continuous random variables.
2. Explain why: $$P(X=x)=0$$ for continuous $X$.
3. Define a PDF.
4. Know that a PDF is a density, not a point probability.
5. Check: $$f(x)\geq0.$$
6. Check: $$\int f(x)\,dx=1.$$
7. Normalize a PDF with an unknown constant.
8. Compute interval probabilities using integration.
9. Use a CDF for continuous variables.
10. Relate: $$F'(x)=f(x).$$
11. Recognize a uniform distribution.
12. Compute uniform interval probabilities quickly.
13. Recognize an exponential waiting-time problem.
14. Use: $$P(X>x)=e^{-\lambda x}.$$
15. Know: $$E[X]=1/\lambda$$ for exponential.
16. Understand the Poisson-count / exponential-waiting-time connection.
17. Understand the memoryless property conceptually.

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Stats 1 is now complete

$$\boxed{ \text{Data description} \rightarrow \text{Association} \rightarrow \text{Counting} \rightarrow \text{Probability} \rightarrow \text{Random variables} \rightarrow \text{Probability distributions} }$$

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Quick Recall & Flashcards

• Week 12 Diagnostic: For a continuous random variable X with PDF f(x), what is P(X = 5)? :: Answer: 0 — For any continuous random variable, the probability of taking any exact single point value is strictly 0: P(X = c) = 0.
• Week 12 Diagnostic: If X is uniformly distributed on [2, 6], what is the value of the PDF f(x) for x in [2, 6]? :: Answer: 0.25 — For X ~ U(a, b), f(x) = 1 / (b - a). Here b = 6, a = 2, so f(x) = 1 / (6 - 2) = 1/4 = 0.25.

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Practice & Self-Check