Question

Let X be the random variable representing the number of calls received in an hour by a 911 emergency service. A probability distribution of X is given below. Value of X 0 1 2 3 4 Probability P(x) 0.32 ____ ____ 0.16 0.08 (a) Suppose the probability that X = 1 and the probability that X = 2 are the same. What are these probabilities? Incorrect: Your answer is incorrect. (b) What is the probability that at least one call received in an hour? (c) What is the expected number of 911 calls in an hour?

Answer 1

Given,

X : the number of calls received in an hour by a 911 emergency service

X	P(x)
0	0.32
1
2
3	0.16
4	0.08

(a) Suppose the probability that X = 1 and the probability that X = 2 are the same.

P(X=1) = P(X=2)

Let 'p' = P(X=1) = P(X=2)

If P(x) to be valid probability function then

i.e

P(X=0)+P(X=1)+P(X=2)+P(X=3) +P(X=4) = 1

Substitute the given values and 'p' for P(X=1) and P(X=2)

0.32 +p+p+0.16+0.08 = 1

2p + 0.56 = 1

2p = 1-0.56=0.44

p = 0.44/2 = 0.22

i.e P(X=1) = P(X=2) = 0.22

X	P(x)
0	0.32
1	0.22
2	0.22
3	0.16
4	0.08

(b)

Probability that at least one call received in an hour = P(X1) = 1 - P(X<1) = 1-P(X=0) = 1-0.32 = 0.68

Probability that at least one call received in an hour = 0.68

(c) the expected number of 911 calls in an hour :

E(X)=expected number of 911 calls in an hour

Formula for E(X)

E(X) = 1.46

The expected number of 911 calls in an hour = 1.46