Question

Give the named probability distribution for each of the following random variables with specific parameter values. You should name the parameters, that is, write the parameter name in your answer (for example, Binomial(n=30,p=0.5))

a) One of Nia’s cat meows when he wants petting. Assuming that the meows are independent of each other and the probability that a meow results in petting is 0.7, what is the distribution of Y: the number of meows until Nia pets him?

(b) Nia has 6 cats. Each cat has a 25% chance of finishing their dinner. The cats eat independently of each other. What is the distribution of X: the total number of cats that finish their dinner?

(c) Nia has a cat that meows a lot. The meows are independent and she meows on average 2 times per minute. What is the distribution of N: number of times this cat meows in half an hour?

(d) There are 30 options of cat toys in a pet store; however, out of those 30, Nia’s cats will only like 4 of them. Nia randomly chooses 2 toys at random and buys them for her cats. What is the distribution of W: the number of toys Nia bought that her cats will like?

Answer 1

Part a)

Nia's cat meows when he wants petting. He will continue to meow until Nia pets him. Hence meow that results in petting is our desired sucess. Probability of sucess is 0.7

Hence, the meows before Nia pets him can be termed as failure.

Y: the number of meows until Nia pets him

We can say that Y is the number of trials to get first sucess.

Hence, Y follows Geometric distribution with parameter p=0.7

Y ~ Geom( p =0.7 )

Part b)

Whether or not each cat finishes it's dinner can be considered as a Bernoulli trial with the probability of success 0.25

Nia has 6 cats. Hence, we can say that the Bernoulli trial is repeated 6 times. Also, all the cats behave independently of each other.

X: the number of cats that finish their dinner

We can say that X is the total number of successes in 6 trials.

Hence, X follows Binomial distribution with the number of trials n=6 and probability of success p= 0.25

X ~ Binomial( n=6, p= 0.25)

Part c)

Nia's cat meows on an average 2 times per minute. We are interested in the number of times this cat meows in half an hour.

We can say that if the cat meows on an average 2 times per minute, then it meows on an average 2*30=60 times in half an hour.

N: the number of times the cat meows in half an hour

Hence, N follows Poisson distribution with average number of meows in half an hour = = 60

N ~ Poisson( = 60 )

Part d)

There are total 30 options of cat toys available in a pet store. Nia is interested in buying only those toys which her cats will like. Hence, the number of toys of our interest is 4. Nia randomly choses 2 toys which she expects her cats will like.

W: the number of toys Nia bought that her cats will like.

We can say that W follows Hypergeometric distribution with population size N= 30 and size of the desired population K= 4 and the number of samples drawn n= 2

Hence, W ~ Hypergeomtric( N=30, K= 4, n=2 )  

I hope you find the solution helpful.

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Thank you in advance!!!