Suppose that the random variable X represents the number of...

Suppose that the random variable $X$ represents the number of heads occurring in three independent tosses of a fair coin. Represent the distribution of $X$ (a) by a listing (b) by a graph of the PMF (c) by a graph of the CDF. Do the same for $Y$, the number of heads occurring in four independent tosses of a fair coin.

Suppose that the random variable $X$ represents the number of heads occurring in three independent tosses of a fair coin. Represent the distribution of $X$
(a) by a listing
(b) by a graph of the PMF
(c) by a graph of the CDF.
Do the same for $Y$, the number of heads occurring in four independent tosses of a fair coin.

Key Concepts

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. In the context of coin tossing, the number of heads (successes) follows a binomial distribution since each toss is independent and has two outcomes (head or tail) with a constant probability of head.

Probability Mass Function (PMF)

The probability mass function for a discrete random variable gives the probability that the variable equals a specific value. For binomial distributions derived from coin tosses, the PMF is calculated using the formula P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where n is the number of tosses, k is the number of heads, and p is the probability of obtaining a head. Graphing the PMF helps visualize the probabilities assigned to each possible outcome.

Cumulative Distribution Function (CDF)

The cumulative distribution function for a discrete random variable is the function that gives the probability that the variable will take a value less than or equal to a given number. It is derived by summing the probabilities from the PMF up to that number. Graphing the CDF provides a visual representation of the accumulation of probabilities and can be used to determine quantiles and assess distribution properties.

Listing the Distribution

Listing the distribution involves enumerating each possible outcome of the random variable along with its corresponding probability. For binomial distributions obtained from coin tosses, this would involve listing outcomes such as the number of heads (e.g., 0, 1, 2, …) and displaying each outcome’s probability calculated using the binomial formula. This method is useful for discrete probability distributions where the total number of outcomes is small.

Transcript

00:01 For this problem, we are told that three fair coins are tossed, and x is the square of the number of heads showing.

00:06 We are first asked to give the probability distribution for the indicated random variable.

00:12 What we'll need to do before that, though, is figure out what our sample space is.

00:17 So we have three fair coins.

00:21 The possible results would be tails, tails, tails, tails, or tails, tails, tails, tails, tails, heads, tails, heads, tails, heads, tails, then there's tails, tails heads heads heads, tails, tails, then there's tails, tails, tails, tails, tails.

00:38 Heads, heads, heads, heads, excuse me, heads, heads, heads, heads, heads, heads, and lastly, tails, or excuse me, heads, heads, heads, rather, not tails, tails, tails, going in order of the number of heads showing.

01:01 We can then see that the values of x corresponding to each would be 0, 1, 4, 4, 4, 4, or excuse me, i need to correct myself here, it is the number of heads, so it would be 0 -1 -1 -1.

01:20 Then we have 4 -4 -4, and then we have a final possibility of 9.

01:28 So our probability distribution table then have the different results.

01:33 We have the probability of each result, where i'll note that we have a total of eight different possible results.

01:41 And we have, or excuse me, not eight different possible results, but we have eight different elements.

01:48 In our sample space is what i'll say.

01:51 Then we have for the probability of getting zero, well, there's a one and eight chance of that happening...

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