Question

discrete probability distributions worksheet

1. you flip four coins. let x, the random variable, be the number of heads on all four coins.

a. list the sample space for the experiment.
b. what are the possible values for x?
c. is the random variable, x, continuous or discrete?
d. construct a probability distribution for this experiment.
x
p(x)
e. construct a histogram for the probability distribution in the space below.

1. determine if the following are probability distributions (if no, state why).

a. x 3 6 9 12 15
p(x) 4/9 2/9 1/9 1/9 1/9
b. x 1 2 3 4 5
p(x) 3/10 1/10 1/10 2/10 3/10
c. x 20 30 40 50
p(x) 1.1 0.2 0.9 0.3

Explanation:

Step1: Check sum of probabilities for part a

For a probability distribution, the sum of all \(P(X)\) values must equal \(1\).

$$ \frac{4}{9}+\frac{2}{9}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}=\frac{4 + 2+1+1+1}{9}=\frac{9}{9}=1 $$

Also, each \(P(X)\geq0\).

Step2: Check sum of probabilities for part b

$$ \frac{3}{10}+\frac{1}{10}+\frac{1}{10}+\frac{2}{10}+\frac{3}{10}=\frac{3 + 1+1+2+3}{10}=\frac{10}{10}=1 $$

And each \(P(X)\geq0\).

Step3: Check sum and individual probabilities for part c

First, check individual probabilities. \(P(X = 20)=1.1>1\). Also,

$$ 1.1+0.2 + 0.9+0.3=2.5 eq1 $$

Answer:

a. It is a probability distribution.
b. It is a probability distribution.
c. It is not a probability distribution because \(P(X = 20)=1.1>1\) (and sum of probabilities \(=2.5
eq1\)).