Let X be a binomial random variable with probability...

Let $X$ be a binomial random variable with probability distribution $$f(x)=\left\{\begin{array}{ll}{ }^{5} C_{x}\left(\frac{1}{3}\right)^{x}\left(\frac{2}{3}\right)^{3-x}, & x=0,1,2, \ldots, 5 \\0, & \text { elsewhere }\end{array}\right.$$ Find the probability distribution of the random variable $Y=X^{3}+2$

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Key Concepts

Transformation of Random Variables

When a new random variable is defined as a function of another, such as Y = g(X), the distribution of the new variable can be derived by mapping each outcome of the original random variable to its corresponding value through the function. For discrete random variables, this typically involves identifying the set of values that the new variable can take and summing or assigning probabilities based directly on the transformation.

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with a constant probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p), and its probability mass function is given by a binomial coefficient multiplied by appropriate powers of p and (1 - p).

Probability Mass Function (PMF)

The probability mass function (PMF) defines the probability that a discrete random variable is exactly equal to some value. In the context of the binomial distribution, the PMF is used to calculate the likelihood of achieving a specific number of successes in a series of trials.

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