Question

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Question 5 (20 pts) You must decide which of two discrete distri- butions a random variable X has. We will call the distributions po and p. Here are the probabilities they assign to the values r of X. 2 Po P 0 1 2 0.1 0.1 0.2 0.1 0.3 0.2 3 4 5 0.3 0.1 0.1 0.1 0.1 0.1 6 0.1 0.1 You have a single observation on X and wish to test Ho: Po is correct H P is correct One possible decision procedure is to reject H, only if X<1. a. Find the probability of a Type I error, that is, the probability that you reject H, when po is the correct distribution. b. Find the probability of a Type II error.

Answer 1

SOLUTION:

Here are the probabilities they assign to the values x of X

X	0	1	2	3	4	5	6
P0	0.1	0.21	02	0.3	0.1	0.1	0.1
P1	0.1	0.3	0.2	0.1	0.1	0.1	0.1

Here H₀: P₀ is correct and

H_a: P₁ is correct.

Also we reject H₀ onlty if x$\leq$1

a)

Here we want to find probability of type-1 error.

i.e., P(reject H₀ when H₀ is true)

$\therefore$ P( x$\leq$1/ H₀ true)

=P( x=0)+P( x=2)

=0.1+0.1=0.2 (probabilities of P₀ i.e., H₀ true)

$\therefore$ probability of type-1 error=0.2

b)

Now,P(type-2 error)

Type-2 error is accept  H₀ is false.

i.e., Accept  H₀ when  H_a is true, (if  H₀ false then H_a true)

$\therefore$P(Accept  H₀/H_a true)=1- P(Reject  H₀/H_a true)

=1-P(x$\leq$1/H_a true)

=1-[P(x=0)+P(x=1)]=1-[0.1+0.3] , (for P₁ as H_a true)

$\therefore$ P(type-2 error)=1-0.4=0.6.