Chapter 4: Q.4.21 (page 174)

Suppose that
$P {X = a} = p, P {X = b} = 1 - p$
(a) show that $\frac{X - b}{a - b}$ is a Bernoulli random variable
(b) Find Var(X).

Short Answer

Expert verified

In the given information the answer os part (a) is $\frac{X - a}{b - a} ~ (\begin{matrix} 0 & 1 \\ p & 1 - p \end{matrix})$ which show that is Bernoulli random variable.

(b) is $Var (X) = (b - a)^{2} p (1 - p)$

Step by step solution

Step 1 :Given Information (Part-a)

With the probability $p$ ,random variable $X$ assumes $α$ .Hence, with the same probability, random variable $\frac{X - a}{b - a}$ assumes $0$

with the probability localid="1646975947088" $1 - p$ ,random variable $X$ assumes $b$ Hence, with the same probability ,random variable $\frac{X - a}{b - a}$ assumes $1$

:Explanation (Part-a)

$\frac{X - a}{b - a} ~ (\begin{matrix} 0 & 1 \\ p & 1 - p \end{matrix})$

which shows that it is a Bernoulli random variable

:Final Answer (Part-a)

The answer is $\frac{X - a}{b - a} ~ (\begin{matrix} 0 & 1 \\ p & 1 - p \end{matrix})$

which shows that $\frac{X - a}{b - a}$ is Bernoulli random variable

:Given Information(Part-b)

We know that $Var (\frac{X - a}{b - a}) = p (1 - p)$ .

Step 5:Explanation (Part-b)

$Var (\frac{X - a}{b - a}) = \frac{1}{(b - a)^{2}} Var (X - a) = \frac{1}{(b - a)^{2}} Var (X)$

so we get that $Var (X) = (b - a)^{2} p (1 - p)$

$Var (X) = (b - a)^{2} p (1 - p)$

:Final Answer (Part-b)

$Var (X) = (b - a)^{2} p (1 - p)$

The answer is $Var (X) = (b - a)^{2} p (1 - p)$ $Var (X) = (b - a)^{2} p (1 - p)$

$Var (X) = (b - a)^{2} p (1 - p)$ $Var (X) = (b - a)^{2} p (1 - p)$

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Suppose that
$P {X = a} = p, P {X = b} = 1 - p$
(a) show that $\frac{X - b}{a - b}$ is a Bernoulli random variable
(b) Find Var(X).

Short Answer

Step by step solution

Step 1 :Given Information (Part-a)

:Explanation (Part-a)

:Final Answer (Part-a)

:Given Information(Part-b)

Step 5:Explanation (Part-b)

:Final Answer (Part-b)

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Suppose thatP{X=a}=p,P{X=b}=1-p(a) show that X-ba-bis a Bernoulli random variable(b) Find Var(X).

Short Answer

Step by step solution

Step 1 :Given Information (Part-a)

:Explanation (Part-a)

:Final Answer (Part-a)

:Given Information(Part-b)

Step 5:Explanation (Part-b)

:Final Answer (Part-b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Decision Maths

Theoretical and Mathematical Physics

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Suppose that
$P {X = a} = p, P {X = b} = 1 - p$
(a) show that $\frac{X - b}{a - b}$ is a Bernoulli random variable
(b) Find Var(X).