Discrete Random Variables and Cumulative Distribution Function, Thesis of Statistics

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Discrete Random Variables

Cumulative Distribution Function Dr. Turgut Tut

What is a random variable?

Would you give an example for a random variable?

Review Question 2

• A shipment of 20 similar laptop computers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives.

Cumulative Distribution Function

• (^) PMF cannot be defined for continuous random variables.
• (^) The cumulative distribution function (CDF) of a random variable is another method to describe the distribution of random variables.
• (^) The advantage of the CDF is that it can be defined for any kind of random variable (discrete, continuous, and mixed). - CDF: FX(x) = P(X≤x), for all x∈ℝ.

Exercis

e

• (^) Toss a coin twice. Let X be the number of observed heads. Find the CDF of X.
• (^) P(0)=P(X=0)=1/4,
• (^) P(1)=P(X=1)=1/2,
• (^) P(2)=P(X=2)=1/4.
• (^) F X(x)^ = P(X≤x) = 0,^ for^ x<0.
• F X (x) = P(X≤x) = 1, for x<=2.

CDF

P(x=1) P(x=2) P(x=0) Always an increasing function

• (^) For all a≤b, we have
  • (^) P(a<X≤b) = F X(b)^ −^ FX(a)
  • P(X≤b) = P(X≤a) + P(a<X≤b)
  • (^) F X(b)^ =^ FX(a)^ +^ P(a<X≤b)
• (^) The CDF gives us P(X≤x). To find P(X<x)
  • P(X<x) = P(X≤x)−P(X=x) = FX(x) − P(x)

Exercis

e

• Let X be a discrete random variable with range R X

Suppose the PMF of X is given by P(k)=1/ k for k=1,2,3,... a. Find and plot the CDF of X, FX(x). b. Find P(2<X≤5) c. Find P(X>4)

CDF

b) c)