(Step-by-Step) If random variable X has pdf. f(x)= \begincasesc x(1-x), 0 Find the constant c. Calculate μ=E(X), σ^2=Var;(X), and P(X ≥qslant

Question: If random variable $X$ has pdf.

\(f(x)= \begin{cases}c x(1-x), & 0

Find the constant c. Calculate $\mu=E(X), \sigma^{2}=\operatorname{Var}(X)$, and $P(X \geqslant 0.65)$.
Find c.d.f $F(x)$ and graph $f(x)$ and $F(x)$.
Find the mode and 75 -th percentile of $X$.
If $X_{1}, X_{2}, \ldots, X_{25}$ is a random sample from this distribution, find $E(\bar{X})$ and $\operatorname{Var}(\bar{X})$, and approximate $P(\bar{X} \geqslant 0.65)$ use normal a approximation
Sketch the graph of the pdf. of the approximate distribution of $\bar{X}$.

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(Step-by-Step) If random variable X has pdf. f(x)= \begincasesc x(1-x), 0 Find the constant c. Calculate μ=E(X), σ^2=Var;(X), and P(X ≥qslant

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