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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 4: Q.4.3 (page 169)

If X has distribution function F, what is the distribution function of the random variable αX + β, where α and β are constants, $α \neq 0 ?$

Short Answer

Expert verified

In the given information the answer is $F_{α X + β} (y) = F (\frac{y - β}{α})$

Step by step solution

01

Step 1:Given Information

X has distribution function F.

The distribution function F at X =x has the property that represents the probability that X is at most y.

$F (x) = P (X \leq x)$

02

Calculation

Let us determine the cumulative distribution function of $α X + β (α \neq 0)$ .

$F_{α X + β} (y) = P (α X + β \leq y)$

$= P (α X \leq y - β)$

$= P (X \leq \frac{y - β}{α})$

$= F (\frac{y - β}{α})$

03

Step 3:Final Answer

The final answer is $F_{α X + β} (y) = F (\frac{y - β}{α})$

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Most popular questions from this chapter

A box contains $5$ red and $5$ blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $\(1.10$ ; if they are different colors, then you win $- \) 1.00$ . (That is, you lose $$ 1.00$ .) Calculate

(a) the expected value of the amount you win;

(b) the variance of the amount you win.

A casino patron will continue to make $\(5$ bets on red in roulette until she has won $4$ of these bets.

What is the probability that she places a total of $9$ bets?
What are her expected winnings when she stops?

Remark: On each bet, she will either win $\) 5$ with probability $\frac{18}{38}$ or lose $$ 5$ with probability $\frac{20}{38}$ .

Let $X$ be a Poisson random variable with parameter $λ$ . Show that $P X = i$ increases monotonically and then decreases monotonically as $i$ increases, reaching its maximum when $i$ is the largest integer not exceeding $λ$ .

Hint: Consider $P X = i / P X = i - 1$ .

Let $X$ be the winnings of a gambler. Let $p (i) =$ $P (X = i)$ and suppose that

$p (0) = 1 / 3; p (1) = p (- 1) = 13 / 55$

$p (2) = p (- 2) = 1 / 11; p (3) = p (- 3) = 1 / 165$

Compute the conditional probability that the gambler wins $i, i = 1, 2, 3,$ given that he wins a positive amount.

The probability of being dealt a full house in a hand of poker is approximately $. 0014$ . Find an approximation for the probability that in $1000$ hands of poker, you will be dealt at least $2$ full houses.

See all solutions

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