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### Transcript of 18.600: Lecture 17 .1in Continuous random sheffield/2019600/Lecture17.pdf Continuous random...

18.600: Lecture 17

Continuous random variables

Scott Sheffield

MIT

Outline

Continuous random variables

Expectation and variance of continuous random variables

Uniform random variable on [0, 1]

Uniform random variable on [α, β]

Measurable sets and a famous paradox

Outline

Continuous random variables

Expectation and variance of continuous random variables

Uniform random variable on [0, 1]

Uniform random variable on [α, β]

Measurable sets and a famous paradox

Continuous random variables

I Say X is a continuous random variable if there exists a probability density function f = fX on R such that P{X ∈ B} =

∫ B f (x)dx :=

∫ 1B(x)f (x)dx .

I We may assume ∫ R f (x)dx =

∫∞ −∞ f (x)dx = 1 and f is

non-negative.

I Probability of interval [a, b] is given by ∫ b a f (x)dx , the area

under f between a and b.

I Probability of any single point is zero.

I Define cumulative distribution function F (a) = FX (a) := P{X < a} = P{X ≤ a} =

∫ a −∞ f (x)dx .

Continuous random variables

I Say X is a continuous random variable if there exists a probability density function f = fX on R such that P{X ∈ B} =

∫ B f (x)dx :=

∫ 1B(x)f (x)dx .

I We may assume ∫ R f (x)dx =

∫∞ −∞ f (x)dx = 1 and f is

non-negative.

I Probability of interval [a, b] is given by ∫ b a f (x)dx , the area

under f between a and b.

I Probability of any single point is zero.

I Define cumulative distribution function F (a) = FX (a) := P{X < a} = P{X ≤ a} =

∫ a −∞ f (x)dx .

Continuous random variables

I Say X is a continuous random variable if there exists a probability density function f = fX on R such that P{X ∈ B} =

∫ B f (x)dx :=

∫ 1B(x)f (x)dx .

I We may assume ∫ R f (x)dx =

∫∞ −∞ f (x)dx = 1 and f is

non-negative.

I Probability of interval [a, b] is given by ∫ b a f (x)dx , the area

under f between a and b.

I Probability of any single point is zero.

I Define cumulative distribution function F (a) = FX (a) := P{X < a} = P{X ≤ a} =

∫ a −∞ f (x)dx .

Continuous random variables

∫ B f (x)dx :=

∫ 1B(x)f (x)dx .

I We may assume ∫ R f (x)dx =

∫∞ −∞ f (x)dx = 1 and f is

non-negative.

I Probability of interval [a, b] is given by ∫ b a f (x)dx , the area

under f between a and b.

I Probability of any single point is zero.

I Define cumulative distribution function F (a) = FX (a) := P{X < a} = P{X ≤ a} =

∫ a −∞ f (x)dx .

Continuous random variables

∫ B f (x)dx :=

∫ 1B(x)f (x)dx .

I We may assume ∫ R f (x)dx =

∫∞ −∞ f (x)dx = 1 and f is

non-negative.

I Probability of interval [a, b] is given by ∫ b a f (x)dx , the area

under f between a and b.

I Probability of any single point is zero.

I Define cumulative distribution function F (a) = FX (a) := P{X < a} = P{X ≤ a} =

∫ a −∞ f (x)dx .

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is P{X ∈ (0, 1) ∪ (3/2, 5)}? I What is F?

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

1 a ≥ 2 I In general P(a ≤ x ≤ b) = F (b)− F (x). I We say that X is uniformly distributed on [0, 2].

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I What is P{X < 3/2}?

I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is P{X ∈ (0, 1) ∪ (3/2, 5)}? I What is F?

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

1 a ≥ 2 I In general P(a ≤ x ≤ b) = F (b)− F (x). I We say that X is uniformly distributed on [0, 2].

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}?

I What is P{1/2 < X < 3/2}? I What is P{X ∈ (0, 1) ∪ (3/2, 5)}? I What is F?

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

1 a ≥ 2 I In general P(a ≤ x ≤ b) = F (b)− F (x). I We say that X is uniformly distributed on [0, 2].

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}?

I What is P{X ∈ (0, 1) ∪ (3/2, 5)}? I What is F?

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is P{X ∈ (0, 1) ∪ (3/2, 5)}?

I What is F?

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is P{X ∈ (0, 1) ∪ (3/2, 5)}? I What is F?

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is P{X ∈ (0, 1) ∪ (3/2, 5)}? I What is F?

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

1 a ≥ 2

I In general P(a ≤ x ≤ b) = F (b)− F (x). I We say that X is uniformly distributed on [0, 2].

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

1 a ≥ 2 I In general P(a ≤ x ≤ b) = F (b)− F (x).

I We say that X is uniformly distributed on [0, 2].

Simple example

I Suppose f (x) =

{ 1/2 x ∈ [0, 2] 0 x 6∈ [0, 2].

I F (a) = FX (a) =

0 a ≤ 0 a/2 0 < a < 2

Another example

I Suppose f (x) =

{ x/2 x ∈ [0, 2] 0 0 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is F?

I FX (a) =

0 a ≤ 0 a2/4 0 < a < 2

1 a ≥ 2

Another example

I Suppose f (x) =

{ x/2 x ∈ [0, 2] 0 0 6∈ [0, 2].

I What is P{X < 3/2}?

I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is F?

I FX (a) =

0 a ≤ 0 a2/4 0 < a < 2

1 a ≥ 2

Another example

I Suppose f (x) =

{ x/2 x ∈ [0, 2] 0 0 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}?

I What is P{1/2 < X < 3/2}? I What is F?

I FX (a) =

0 a ≤ 0 a2/4 0 < a < 2

1 a ≥ 2

Another example

I Suppose f (x) =

{ x/2 x ∈ [0, 2] 0 0 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}?

I What is F?

I FX (a) =

0 a ≤ 0 a2/4 0 < a < 2

1 a ≥ 2

Another example

I Suppose f (x) =

{ x/2 x ∈ [0, 2] 0 0 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is F?

I FX (a) =

0 a ≤ 0 a2/4 0 < a < 2

1 a ≥ 2

Another example

I Suppose f (x) =

{ x/2 x ∈ [0, 2] 0 0 6∈ [0, 2].

I What is P{X < 3/2}? I What is P{X = 3/2}? I What is P{1/2 < X < 3/2}? I What is F?

I FX (a) =

0 a ≤ 0 a2/4 0 < a < 2

1 a ≥ 2

Outline

Continuous random variables

Expectation and variance of continuous random variables

Uniform random variable on [0, 1]

Uniform random variable on [α, β]

Measurab

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