Let X be a binomial random variable with prob ability...

Let $X$ be a binomial random variable with prob ability distribution $$ f(x)=\left\{\begin{array}{l}\left(\begin{array}{l}3 \\x\end{array}\right)\left(\frac{2}{5}\right)^{x}\left(\frac{3}{5}\right)^{3-x} \\0\end{array}\right.$$ $$x=0,1,2,3$$ Find the probability distribution of the random variable $Y=X^{2}$.

Let $X$ be a binomial random variable with prob ability distribution $$
f(x)=\left\{\begin{array}{l}\left(\begin{array}{l}3 \\x\end{array}\right)\left(\frac{2}{5}\right)^{x}\left(\frac{3}{5}\right)^{3-x} \\0\end{array}\right.$$ $$x=0,1,2,3$$ Find the probability distribution of the random variable $Y=X^{2}$.

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Key Concepts

Binomial Distribution

The binomial distribution describes the probability of obtaining a fixed number of successes in a fixed number of independent trials, with each trial having the same probability of success. It is characterized by two parameters: the number of trials and the probability of success in a single trial.

Discrete Random Variable Transformation

When a function is applied to a discrete random variable, the probability distribution is transformed accordingly. This involves mapping the original outcomes to new outcomes and summing the probabilities of all original outcomes that correspond to the same transformed value.

Probability Mass Function (PMF)

The probability mass function gives the probability that a discrete random variable is exactly equal to some value. When the random variable is transformed using a function, the PMF of the new random variable is determined by summing the probabilities of the original variable for all values that map to the same new value.

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