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What is a geometric random variable and what are its possible values?
The geometric random variable is used when one is modelling a series of experiments that have one of two possible outcomes – sucess or failure. The geometric random variable tells you the number of experiments that were performed before obtaining a sucess. This random variable can thus take values of 1, 2, 3.
How do you know if a random variable is geometric?
The random variable is defined as X = number of trials UNTIL a 3 occurs. To VERIFY that this is a geometric setting, note that rolling a 3 will represent a success, and rolling any other number will represent a failure. The probability of rolling a 3 on each roll is the same: 1/6. The observations are independent.
What’s the expected value of the geometric random variable?
It depends on how you’ve set up the geometric random variable. Here, Sal is setting X to be the number of trials you need before you get a successful outcome. Your teacher, on the other hand, set X to be the number of failures before the first successful outcome.
Is the expectation of a random variable a linear operator?
In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].
How to calculate the probability of a geometric variable?
Let X ∼ Geom(p). Then Where Σ1 = ∑∞n = 1n(1 − p)n − 1. Then let Σ0 = 1 + (1 − p) + (1 − p)2 + … = 1 1 − 1 + p = 1 p as we have a geometric series.
What is the expectation of a Cauchy random variable?
A Cauchy random variable takes a value in (−∞,∞) with the fol- lowing symmetric and bell-shaped density function. f(x) = 1 π[1+(x−µ)2] The expectation of Bernoulli random variable implies that since an indicator function of a random variable is a Bernoulli random variable, its expectation equals the probability.