Continuous Probability Distributions

Continuous vs Discrete Distributions

Aspect	Discrete	Continuous
Values	Countable (0, 1, 2, …)	Uncountable (any value in range)
Function	PMF: P(X = x)	PDF: f(x)
P(X = specific value)	Can be positive	Always 0
Probabilities	Sum up	Integrate (area under curve)

Probability Density Function (PDF)

The probability density function f(x) describes the relative likelihood of values. Probability = area under the curve.

PDF Properties

$f(x) \geq 0$ for all x
Total area under curve = 1
$P(a \leq X \leq b) = \int_a^b f(x) \, dx$ (area between a and b)

Mean and Variance (Continuous)

$\mu = E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$

$\sigma^2 = Var(X) = \int_{-\infty}^{\infty} (x-\mu)^2 \cdot f(x) \, dx$

1. Uniform Distribution

The uniform distribution assigns equal probability to all values in an interval [a, b].

Continuous Uniform Distribution

$X \sim \text{Uniform}(a, b)$

$f(x) = \frac{1}{b-a} \quad \text{for } a \leq x \leq b$

Mean: $\mu = \frac{a+b}{2}$

Variance: $\sigma^2 = \frac{(b-a)^2}{12}$

Random Number Generator

A random number is generated uniformly between 0 and 10.

X ~ Uniform(0, 10)

P(X < 3)? $P(X < 3) = \frac{3-0}{10-0} = 0.3$

P(2 < X < 7)? $P(2 < X < 7) = \frac{7-2}{10-0} = 0.5$

Mean and SD:

μ = (0+10)/2 = 5
σ² = (10-0)²/12 = 100/12 = 8.33
σ = 2.89

Waiting for a Bus

Buses arrive every 15 minutes. You arrive at a random time.

Wait time X ~ Uniform(0, 15)

P(wait less than 5 minutes)? $P(X < 5) = \frac{5}{15} = \frac{1}{3} = 0.333$

Expected wait time: $\mu = \frac{0+15}{2} = 7.5 \text{ minutes}$

2. Exponential Distribution

The exponential distribution models time between events in a Poisson process. It’s the continuous counterpart to the geometric distribution.

Exponential Distribution

$X \sim \text{Exponential}(\lambda)$

$f(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0$

Mean: $\mu = \frac{1}{\lambda}$

Variance: $\sigma^2 = \frac{1}{\lambda^2}$

CDF: $P(X \leq x) = 1 - e^{-\lambda x}$

Customer Arrivals

Customers arrive at rate λ = 4 per hour (Poisson process).

Time between arrivals X ~ Exponential(4)

Mean time between customers: $\mu = \frac{1}{4} = 0.25 \text{ hours} = 15 \text{ minutes}$

P(next customer arrives within 10 minutes = 1/6 hour)? $P(X \leq 1/6) = 1 - e^{-4 \times 1/6} = 1 - e^{-0.667} = 1 - 0.513 = 0.487$

P(wait more than 30 minutes = 0.5 hours)? $P(X > 0.5) = e^{-4 \times 0.5} = e^{-2} = 0.135$

Memoryless Property

The exponential distribution is memoryless: the probability of waiting t more time units doesn’t depend on how long you’ve already waited.

$P(X > s + t \,|\, X > s) = P(X > t)$

Memoryless Property

Light bulb lifetime ~ Exponential with mean 1000 hours.

If a bulb has lasted 500 hours, P(lasting 200 more hours)?

Same as P(new bulb lasting 200 hours): $P(X > 200) = e^{-200/1000} = e^{-0.2} = 0.819$

The bulb doesn’t “remember” it’s been on for 500 hours!

3. Chi-Square Distribution

The chi-square distribution arises when you sum squared standard normal variables. It’s crucial for hypothesis testing.

Chi-Square Distribution

If $Z_1, Z_2, \ldots, Z_k$ are independent standard normal variables:

$X = Z_1^2 + Z_2^2 + \cdots + Z_k^2 \sim \chi^2(k)$

Mean: $\mu = k$ (degrees of freedom)

Variance: $\sigma^2 = 2k$

Properties

Only takes positive values (x ≥ 0)
Right-skewed (especially for small df)
Approaches normal as df increases
Used in chi-square tests, confidence intervals for variance

Chi-Square Critical Values

Common critical values for chi-square tests:

df	χ²₀.₉₅ (right tail = 0.05)	χ²₀.₀₅ (left tail = 0.05)
1	3.84	0.004
5	11.07	1.15
10	18.31	3.94
20	31.41	10.85

4. t-Distribution

The t-distribution (Student’s t) is similar to the normal but with heavier tails. It’s used when estimating means with unknown population standard deviation.

t-Distribution

$T \sim t(df)$

Mean: μ = 0 (for df > 1)

Variance: $\sigma^2 = \frac{df}{df-2}$ (for df > 2)

Properties

Symmetric and bell-shaped like normal
Heavier tails than normal (more probability in extremes)
Approaches standard normal as df → ∞
At df ≈ 30, very close to normal

t vs z Critical Values

Confidence	z (normal)	t (df=10)	t (df=30)
90%	1.645	1.812	1.697
95%	1.96	2.228	2.042
99%	2.576	3.169	2.750

With small samples, you need to go further from the mean for the same confidence!

5. F-Distribution

The F-distribution is the ratio of two chi-square variables. It’s used in ANOVA and comparing variances.

F-Distribution

If $X_1 \sim \chi^2(df_1)$ and $X_2 \sim \chi^2(df_2)$ are independent:

$F = \frac{X_1/df_1}{X_2/df_2} \sim F(df_1, df_2)$

Mean: $\mu = \frac{df_2}{df_2-2}$ (for df₂ > 2)

Properties

Only takes positive values
Right-skewed
Has TWO degrees of freedom parameters
Used in F-tests, ANOVA, regression analysis

6. Beta Distribution

The beta distribution is flexible for modeling proportions and probabilities.

Beta Distribution

$X \sim \text{Beta}(\alpha, \beta)$

Values in [0, 1]

Mean: $\mu = \frac{\alpha}{\alpha + \beta}$

Different α and β values create different shapes:

α = β = 1: Uniform distribution
α = β > 1: Symmetric, bell-shaped
α > β: Left-skewed
α < β: Right-skewed

Comparison of Continuous Distributions

Distribution	Support	Parameters	Common Use
Normal	(-∞, ∞)	μ, σ	Natural phenomena
Uniform	[a, b]	a, b	Random number generation
Exponential	[0, ∞)	λ	Wait times
Chi-square	[0, ∞)	df	Variance tests
t	(-∞, ∞)	df	Small sample means
F	[0, ∞)	df₁, df₂	ANOVA, variance comparison
Beta	[0, 1]	α, β	Proportions

Choosing the Right Distribution

Identifying Distributions

Scenario 1: Time until next earthquake → Exponential (waiting time for rare event)

Scenario 2: Test scores of large population → Normal (central limit theorem)

Scenario 3: Bus arrival time within a 10-minute window → Uniform (any time equally likely)

Scenario 4: Sample variance of normal data → Chi-square (sum of squared normals)

Scenario 5: Batting average, proportion successful → Beta (values between 0 and 1)

Relationships Between Distributions

Many distributions are related:

Normal → Chi-square: $Z^2 \sim \chi^2(1)$
Chi-square + Chi-square → F: Ratio of chi-squares
Normal + Chi-square → t: Z divided by sqrt(χ²/df)
Poisson → Exponential: Time between Poisson events
Binomial → Normal: For large n (CLT)
Exponential → Gamma: Sum of exponentials

Summary

In this lesson, you learned:

Continuous distributions use PDFs; probability = area under curve
Uniform: Equal probability over an interval
Exponential: Time between events (memoryless)
Chi-square: Sum of squared normals (variance tests)
t-distribution: Like normal but heavier tails (small samples)
F-distribution: Ratio of chi-squares (ANOVA)
Beta: Flexible distribution for proportions
Distributions are interconnected through mathematical relationships

Practice Problems

1. X ~ Uniform(5, 15). Find: a) P(X < 8) b) P(7 < X < 12) c) Mean and standard deviation

2. Phone calls arrive at rate 2 per hour (Poisson). Time between calls is exponential. a) Mean time between calls? b) P(more than 1 hour between calls)? c) P(call arrives in next 15 minutes)?

3. A chi-square distribution has df = 8. What is: a) The mean? b) The variance?

4. For a t-distribution with df = 15, the critical value for a two-tailed test at α = 0.05 is 2.131. What does this mean?

Click to see answers

1. a) P(X < 8) = (8-5)/(15-5) = 3/10 = 0.3 b) P(7 < X < 12) = (12-7)/(15-5) = 5/10 = 0.5 c) μ = (5+15)/2 = 10 σ² = (15-5)²/12 = 100/12 = 8.33 σ = 2.89

2. λ = 2 per hour a) Mean = 1/λ = 0.5 hours = 30 minutes b) P(X > 1) = e^(-2×1) = e^(-2) ≈ 0.135 c) P(X < 0.25) = 1 - e^(-2×0.25) = 1 - e^(-0.5) = 1 - 0.607 ≈ 0.393

3. a) Mean = df = 8 b) Variance = 2×df = 2×8 = 16

4. If the test statistic |t| > 2.131, we reject the null hypothesis. This leaves 2.5% probability in each tail of the distribution.

Next Steps

Continue building your statistical knowledge:

Sampling Distributions - Distributions of sample statistics
Confidence Intervals - Using distributions for estimation
T-Test Calculator - Apply t-distribution to real tests

Continuous vs Discrete Distributions

Probability Density Function (PDF)

1. Uniform Distribution

2. Exponential Distribution

Memoryless Property

3. Chi-Square Distribution

Properties

4. t-Distribution

Properties

5. F-Distribution

Properties

6. Beta Distribution

Comparison of Continuous Distributions

Choosing the Right Distribution

Relationships Between Distributions

Summary

Practice Problems

Next Steps

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