If X be a random variable taking values `x_(1),x_(2),x_(3),….,x_(n)` with probabilities `P_(1),P_(2),P_(3)`,…..`P_(n)`, respectively. Then, Var (x) is equal to …….. .
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A box has 5 blue and 4 red balls. One ball is drawn at random and replaced. Its colour is also not noted. Then, another ball is drawn at random. What
A box = {5 blue, 4 red} Let `E_(1)` is the event that first ball drawn is blue, `E_(2)` is the event that first ball drawn is red and E...
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A sample space consists of 3 sample points with associated probabilities given as `2p ,p^2,4p-1.` Then the value of `p` is
Correct Answer - A `P^(2) + 2p + 4p - 1 = 1` (Exhaustive) `P^(2) + 6p - 2 = 0` or `p = - 3 pm sqrt(11)` `= sqrt(11) -...
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Identify the random variable as either discrete or continuous in each of the following. If the random variable is discrete,
(i) Let X = number of unemployed graduates in a town. Since the population of the town is 1 lakh, X takes the finite values. ∴ random variable X is discrete. Range =...
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Evaluating integrals dependent on a parameter: Differentiate I with respect to the parameter with in the sign an integrals taking variable of the inte
Correct Answer - B Let `I(a)=int_(0)^(1)(x^(a)-1)/(logx)dx`……………..1 Differentiating w.r.t `a` keeping `x` is constant, we get `(dI(a))/(da)=int_(0)^(1)((x^(a)-1)/(logx))dx` `=int_(0)^(1)(x^(a)logx)/(logx)dx` `=int_(0)^(1)x^(a)dx` `=(x^(a+1))/(a+1)|_(0)^(1)=1/((a+1))` Integrating both sides w.r.t `a` we get `I(a)=log(a+1)+c` For `a=0,1(0)=log1+c` [from equation 1]...
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`A B C` is a variable triangle such that `A` is (1, 2), and `Ba n dC` on the line `y=x+lambda(lambda` is a variable). Then the locus of the orthocentr
Correct Answer - D As the altitude from A is fixed and the orthocenter lies on the altitude, x+y=3 is the required locus.
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If the coordinates of a variable point `P` are `(acostheta,bsintheta),` where `theta` is a variable quantity, then find the locus of `Pdot`
Let `P -=(x,y)`. According to the question, `x=acostheta`(1) `y=bsintheta` (2) Sqaureing and adding (1) and (2), we get `(x^2)/(a^2)+(y^2)/(b^2)=cos^2theta+sin^2theta` or `(x^2)/(a^2)+(y^2)/(b^2)=1`
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Suppose families always have one, two, or three children, with probabilities 1/4, 1/2, and 1/4, respectively. Assume everyone eventually gets married
Let the events be A: exactly one child B : exactly two children C : exactly three children E : couple has exactly 4 grandchildren `thereforeP(A)=1/4,P(B)=1/2,P(C)=1/4` `thereforeP(E)=P(A).P(E//A)+P(B).P(E//B)+P(C).P(E//C)` Now when couple...
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Events `Aa n dC` are independent. If the probabilities relating `A ,B ,a n dC` are `P(A)=1//5, P(B)=1//6; P(AnnC)=1//20 ; P(BuuC)=3//8.` Then events `
Correct Answer - A `P(AnnC)=P(A)P(C)` `or1/20=1/5P(C)` `orP(C)=1/4` Now, `P(BuuC)=1/6+1/4-P(BnnC)` `P(BnnC)=5/12-3/8=1/24P(BnnC)` Therefore, B and C aer independent.
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Whenever horses `a , b , c` race together, their respective probabilities of winning the race are 0.3, 0.5, and 0.2 respectively. If they race three t
Correct Answer - A `P(a)=0.3P(b)=0.5,P(c)=0.2.` Hence, a,b c are exhaustive. P(same horse wins all the three reces) =P(aaa or bbb or ccc) `=(0.3)^(3)+(0.5)^(3)+(0.2)^(3)` `=(27=125+8)/(1000)=(160)/(1000)=4/25` P(each horse wins exactly one race) =P...
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If l and m are variable real number such that `5l^(2)+6m^(2)-4lm+3l=0`, then the variable line lx+my=1 always touches a fixed parabola, whose axes is
Correct Answer - B (2) Any parabola whose axes whose axes is parallel to the x-axis will be of the from `(y-a)^(2)=4b(x-c)` (1) Now, lx+my=1 can be rewritten as `y-a=-(l)/(m)(x-c)+(1-am-lc)/(m)` (2)...
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