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Two cards drawn at random from a pack of cards with replacement. Let the random variable x be the number of card drawn from the heart suit.

AConstruct a probability mass function
BConstruct a cumulative distribution function
CWhat is the most likely value of the random variable X?

Verified Answer :

LammettHash

This is a discrete distribution with three possible cases of interest. In any such two-card hand, you can have (1) no cards, (2) one card, or (3) both cards with the heart suit.

\(\mathbb P(X=x)=\begin{cases}\dfrac{\dbinom{39}2}{\dbinom{52}2}=\dfrac{19}{34}&\text{for }x=0\\\\\dfrac{\dbinom{39}1\dbinom{13}1}{\dbinom{52}2}=\dfrac{13}{34}&\text{for }x=1\\\\\dfrac{\dbinom{13}2}{\dbinom{52}2}=\dfrac1{17}&\text{for }x=2\end{cases}\)

As a PMF, you would say

\(f_X(x)=\begin{cases}\dfrac{19}{34}&\text{for }x=0\\\\\dfrac{13}{34}&\text{for }x=1\\\\\dfrac1{17}&\text{for }x=2\\\\0&\text{otherwise}\end{cases}\)

The CDF is given by

\(\displaystyle F_X(x)=\begin{cases}0&\text{for }x<0\\\\\dfrac{19}{34}&\text{for }0\le x<1\\\\\dfrac{32}{34}&\text{for }1\le x<2\\\\1&\text{for }x\ge2\end{cases}\)

The most likely value of \(X\) is the expected value for the distribution, which is given by

\(\displaystyle\mathbb E(X)=\sum_xxf_X(x)=0\times f_X(0)+1\times f_X(1)+2\times f_X(2)=\dfrac{17}{34}=\dfrac12\)

Probability mass function is probability function evaluated for discrete distributions. The pmf, cdf, and expectation of random variable X are:

  • PMF(probability mass function):

\(P(X = 0) = P(B)P(B) =\dfrac{1}{16}\\\\P(X = 1) = 2P(A)P(B) =\dfrac{6}{16}\\\\P(X = 2) = P(A)P(A) =\dfrac{9}{16}\)

  • (and
  • probability
  • is 0 for any other value of X)
  • CDF(Cumulative distributive function):For X < 0, F(X) = 0
  • \(F(X = 0) = P(X \leq 0) = P(X = 0) = \dfrac{1}{16}\\\\F(X = 1) = P(X \leq 1) = P(X = 0) + P(X = 1) = \dfrac{7}{16}\\\\F(X = 2) = P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = \dfrac{16}{16} = 1\)
  • For X > 2, F(X) = 1
  • Most likely value of random variable X:

What is the most likely value of a random variable?

It is its expected value, denoted by E(X).

How to calculate the expectation of a discrete random variable?

Expectation can be taken as a weighted mean, weights being the probability of occurrence of that specific observation.

Thus, if the random variable is X, and its probability mass function is given to be f(x) = P(X = x), then we have:

\(E(X) = \sum_{i=1}^n( f(x_i) \times x_i)\)

(n is number of values X takes)

What is probability mass function of a random variable X?

PMF (probability mass function) is a function describing probabilities for each value of discrete random variable X.

What is cumulative distribution function?

Suppose that for a random variable X, its probability function be f(x).

Then we have:

\(CDF = F(x) = f(X\leq x)\)

For the given case, X is tracking the number of cards drawn belonging to heart suit.

Since there are only 2 cards drawn, thus, there can be at max 2 heart suit cards and at minimum, can be 0 heart suit cards.

Let two events be:

A = drawing a card belonging to heart suit

B = drawing a card not belonging to heart suit

Thus,

\(P(A) = \dfrac{^{13}C_1}{^{52}C_1} = \dfrac{13}{52} = \dfrac{1}{4}\)

\(P(B) = \dfrac{^{39}C_1}{^{52}{C_1}} = \dfrac{39}{52} = \dfrac{3}{4}\)

Then, as we're doing replacement, thus, using chain rule, we get:

\(P(X = 0) = P(B)P(B) =\dfrac{1}{16}\\\\P(X = 1) = 2P(A)P(B) =\dfrac{6}{16}\\\\P(X = 2) = P(A)P(A) =\dfrac{9}{16}\)

(and probability is 0 for any other value of X)

P(X = 1) = 2P(A)P(B) since either first card can be heart or second can be heart. Thus, 2 orders.

For cumulative distribution function, we get:

For X < 0, F(X) = 0

\(F(X = 0) = P(X \leq 0) = P(X = 0) = \dfrac{1}{16}\\\\F(X = 1) = P(X \leq 1) = P(X = 0) + P(X = 1) = \dfrac{7}{16}\\\\F(X = 2) = P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = \dfrac{16}{16} = 1\)

For X > 2, F(X) = 1

For getting most likely value of X, we get:

\(E(X) = 0.P(X=0) + 1.P(X=1) + 2.P(X=2) = \dfrac{6}{16} + \dfrac{18}{16} = 1.5\)

Learn more about cumulative distribution function here:

https://brainly.com/question/19884447

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