Zadanie 2d
Show that .
Random-variable moments and densities
Analyse discrete or continuous random variables, calculate moments and quantiles, and track effects of linear transformations.
Tę umiejętność sprawdza 50 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 50 Otwórz w wyszukiwarce z filtrami
Zadanie 2e
On the branch moving in the positive direction, write the positive velocity as v₊(s). For −A<s<A, the density has form fS(s) = k/v+(s) with k>0. Find k.
Zadanie 2f
(i) Find E(S), with justification. (ii) Interpret this value in the scanning-head model.
Zadanie 6a
Find a exactly.
Zadanie 6b
Find Var(X).
Zadanie 7a
Show that b = 1/3.
- Materiał graficzny
Zadanie 7b
Given E(X) = 4.8, find a.
Zadanie 7a
Find P(0.5≤X<1.5).
Zadanie 7b
Find (i) b; (ii) a,c.
Zadanie 11a
(i) Evaluate . (ii) Write down the mode of T. (iii) Determine whether the median is larger or smaller than the mode, with a reason.
Zadanie 11b
A cycle is classified as short if it lasts no more than 6 minutes. Find the probability that a randomly selected cycle is short.
Zadanie 11c
A cycle is classified as short when T ≤ 6. Given that a selected cycle is short, find the probability that it lasts no more than 5.5 minutes.
Zadanie 11d
Find the lower quartile of T.
Zadanie 11e
A performance score is S = a − bT, with a, b > 0. Its mean is 120 and its maximum possible value is 180. Using E(T) = 7.59472, find a and b, each rounded to the nearest integer.
Zadanie 11f
The calibration uses S = a − bT, with a, b > 0, mean score 120 and maximum possible score 180. Use E(T) = 7.59472 and round the resulting coefficients to the nearest integer before calculating the score variance. The score is S = a − bT, with the coefficients…
Zadanie 11b
Show that for a > 0, .
Zadanie 11c
(i) Use l’Hôpital’s rule to evaluate . (ii) Hence use the density condition to find k.
Zadanie 11d
Find the median total measurement time.
Zadanie 7b
For a=0.1, find Var(3−2X).
Zadanie 11b
(i) Show that . (ii) Find k to six significant figures.
Zadanie 11c
Find (i) E(X); (ii) Var(X).
Zadanie 11d
Writing μ=E(X) and σ²=Var(X), find P(μ−σ<X<μ+σ).
Zadanie 11a
Show that 625a/4 + 25b/2 = 1.
Zadanie 11b
(i) Show that 25a+b=0. (ii) Find a and b.
Zadanie 11c
Show that the median lifetime is smaller than the mode.
Zadanie 11d
A coating has already lasted more than 6 days. According to the model, what is the probability that it lasts at least 6.6 days?
Zadanie 1d
For policy A with p=.12, find E(X) and Var(X).
Zadanie 1g
The distribution of Y is P(Y=1)=1/16,P(Y=2)=m,P(Y=3)=405/1024,P(Y=4)=n. (i) Find m,n. (ii) Show that E(Y)=3019/1024. (iii) Find Var(Y).
Zadanie 1h
(i) Using E(X)=1/p, find p giving E(X)=E(Y). (ii) Find Var(X) for that p. (iii) Which policy gives more consistent inspection intervals at this common mean? Give a reason.
Zadanie 8a
Show that 5k³=6.
Zadanie 8b
Find the median.
Zadanie 10a
Find the mode of X.
Zadanie 10b
Find P(1≤X≤2).
Zadanie 10c
Find the median of X.
Zadanie 10d
Find the probability that a randomly selected customer spends at most USD 32.
Zadanie 10e
Find the expected amount spent per customer, to the nearest cent.
Zadanie 6a
Find E(X).
Zadanie 6b
If E(c−3X)=0, find the constant c.
Zadanie 6c
Find the median of X.
Zadanie 6a
State E(X) in terms of a.
Zadanie 6b
Use integration to find Var(X).
Zadanie 11a
Find E(X) exactly.
Zadanie 11b
Find P(X<0.6).
Zadanie 6b
Find Var(Y).
Zadanie 6b
Find Var(Y).
Zadanie 7a
Find k.
Zadanie 7b
Find E(X).
Zadanie 8
A continuous random variable X has density for a≤x≤c, for c<x≤b, and 0 elsewhere. Assume a<c<b and c ≥ (2b + a)/3. Find its median in terms of a,b,c.
Zadanie 6a
Express a in terms of b.
Zadanie 6b
For b=2 and a = 1/(1 − 3e−2), find the median.