How to calculate z score in r programming

Answer

What is Lambda (λ) in Statistics?

In statistics, lambda (λ) is a parameter used in various contexts to represent a rate, mean, or regularization constant depending on the statistical model. It appears frequently in probability distributions and inferential statistics, most notably in the Poisson and Exponential distributions.

1. Lambda (λ) as a Rate Parameter

(a) Poisson Distribution

The Poisson distribution models the number of occurrences of an event within a fixed interval of time or space when these events occur independently at a constant average rate.

The parameter λ represents both:

• The average number of events in the interval, and

• The rate parameter of the distribution.

The probability mass function (PMF) is:

$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!},k=0,1,2,\dots$

$P(X$

• Mean: $E[X] = \lambda$

• Variance: $Var(X) = \lambda$

Example:
If an emergency room receives an average of 3 patients per hour, then λ = 3.
The probability of exactly 5 patients arriving in an hour is:

$P(X=5)=\frac{e^{-3}\cdot 3^5}{5!}=0.1008$

(b) Exponential Distribution

The Exponential distribution models the time between events in a Poisson process.
Here, λ represents the rate of occurrence per unit time.

The probability density function (PDF) is:

$f(x;\lambda )=\lambda e^{-\lambda x},x\ge 0$

• Mean: $E[X] = \frac{1}{\lambda}$

• Variance: $Var(X) = \frac{1}{\lambda^2}$

Example:
If calls arrive at a help desk at a rate of 2 per minute, then λ = 2.
The probability that the next call comes after more than 1 minute is:

$P(X>1)=e^{-2(1)}=e^{-2}=0.1353$

2. Lambda (λ) as a Regularization Parameter

In regression analysis, particularly in Ridge and Lasso regression, λ controls the penalty applied to large coefficients.

(a) Ridge Regression

The Ridge objective function is:

Yes · 0 No · 0 Add Comment · 0

Add ·

Report